Matrix and Determinant

Matrix-Originated by Chinese mathematician around 200 AD. However Determinant-Originated by Japanese mathematician around 600BC. Formal origin-at the age of Gauss
In general, Matrix is arrangement of mathematical objects in terms of row and column.
It can be used to

1. describe two-variable phenomenon using cross-tabs
2. projec of three-dimensional image into a two-dimensional screen
3. express sequence of numbers in a binary format

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Definition of Matrix

A matrix is rectangular arrangement of numbers arranged into a fixed number of rows and columns.

Matrices are usually named using upper case letters. For example,
\( A=\begin{bmatrix} 1&2&3\\4&5&6\\7&8&9\end{bmatrix} \) is a matrix.

In this example, each element in the arrangement in A is called an entry or an element of the matrix A. The elements are written based on the place value system as
\( A=\begin{bmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23} \\a_{31}&a_{32}&a_{33} \end{bmatrix}\)

For example,
\( \begin{bmatrix} a_{11}& a_{12} &a_{13} \end{bmatrix}\) is first row of A
\( \begin{bmatrix} a_{11}\\a_{21}\\a_{31} \end{bmatrix}\) is first column of A.

यहाँ सदस्यहरूलाई a₁₁,a₁₂,a₁₃,... रूपमा प्रस्तुत गर्दा मेट्रिक्सका कुनै पनि सदस्यको पङक्ति र लहर सजिलै पत्ता लगाउन सकिन्छ
a ₁₁ पहिलो पङ्क्ति र पहिलो लहरमा पर्छ ।
a₂₁ दोस्रो पङ्क्ति र पहिलो लहरमा पर्छ, त्यस्तै
a₂₃ दोस्रो पङ्क्ति र तेस्रो लहरमा पर्छ ।
यसरी मेट्रिक्सका सदस्यहरूलाई a_mnका रूपमा लेख्ने गरिन्छ, जहाँ m पङ्क्ति र n लहरको सङ्ख्या हो ।

Note

1. The individual numbers in a matrix are called the elements.
2. A horizontal arrangement of the numbers in a matrix is called a row.
3. A vertical arrangement of the numbers in a matrix is called a column.
4. The number of rows and the number of columns in a matrix is called the dimensions of a matrix.

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Order and Size of Matrix

The number of rows and columns in a Matrix is called order of the Matrix. It is also called size, index, or dimension of the Matrix.

If a matrix A has m rows and n column then order of A is written as m x n and read as m by n.

If a matrix A is given by
\( A=\begin{bmatrix} 1&2&3\\4&5&6\\7&8&9\end{bmatrix} \)
Then, the order of a matrix A is
3 by 3.

Example 1

Construct a 2 x 2 Matrix A where \( a_{ij} =\frac{(i+j)^2}{2}\)

Solution
We know that, the 2 x 2 Matrix is of the form
\( A=\begin{bmatrix} a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}\)
Here \( a_{ij} =\frac{(i+j)^2}{2}\), therefore, the elements are
\( a_{11}=\frac{(1+1)^2}{2}= 2\)
\( a_{12}=\frac{(1+2)^2}{2}=9/2 \)
\( a_{21}=\frac{(2+1)^2}{2}=9/2 \)
\( a_{22}=\frac{(2+2)^2}{2}=8 \)
Therefore, the required Matrix is
\( A=\begin{bmatrix} 2&9/2\\9/2&8\end{bmatrix}\)

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Types of Matrix

1. Singleton Matrix
   A matrix with only one element is called Singleton Matrix.
   For example,
   A=[23].
   The order of such matrix is 1 x 1.
2. Row matrix
   A row matrix has only one row but any number of columns. A matrix is said to be a row matrix if it has only one row.
   For example,
   \(A=\begin{bmatrix} -1&2&√5&23\end{bmatrix} \)
   is a row matrix of order 1 x 4 .
   In general, row matrix has order 1 x n.
3. Column Matrix
   A column matrix has only one column but any number of rows. A matrix is said to be a column matrix if it has only one column.
   For example,
   \( A=\begin{bmatrix} 0 \\ 3 \\-11 \\ 2 \end{bmatrix} \)
   is a column matrix of order 4 x 1.
   In general, column matrix has order m x 1.
4. Rectangular Matrix
   A matrix is said to be a rectangular matrix if the number of rows is not equal to the number of columns.
   For example,
   \( A=\begin{bmatrix} 33&2&47\\2&-1&√3\\2&3&2\\1&-1&-5\end{bmatrix} \)
   is a matrix of the order 4 x 3
5. Square matrix
   A square matrix has the number of columns equal to the number of rows. A matrix in which the number of rows is equal to the number of columns is said to be a square matrix.
   Thus an m x n matrix is said to be a square matrix if m = n and is known as a square matrix of order n.
   For example,
   \( A=\begin{bmatrix} 33&2&4\\1&3&23\end{bmatrix} \)
   is a square matrix of order 3.
   In general, square matrix has order m.
6. Diagonal Matrix
   A square matrix \( B = \begin{bmatrix} bij \end{bmatrix}_{m \times m}\) is said to be a diagonal matrix if all its non-diagonal elements are zero, that is a matrix \( B =\begin{bmatrix} bij\end{bmatrix}_{m\times m}\) is said to be a diagonal matrix if \( bij = 0\) , when \( i \ne j\) .
   For example,
   \( A=\begin{bmatrix} 4 \end{bmatrix} \)
   \( B=\begin{bmatrix}-1&0\\0&2\end{bmatrix} \)
   \( C=\begin{bmatrix} 3&0&0\\0&-5&0\\0&0&2\end{bmatrix}\)
   are diagonal matrices of order 1, 2, 3 respectively.
7. Scalar Matrix
   A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant. That is, a diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is,
   a square matrix \( B = \begin{bmatrix} bij\end{bmatrix}_{n \times n}\) is said to be a scalar matrix if \( bij = 0\), when \( i \ne j\) and \( bij = k\) , when \( i = j\) , for some constant k .
   For example,
   \( A=\begin{bmatrix} 4\end{bmatrix} \)
   \( B= \begin{bmatrix} -1&0\\0&-1\end{bmatrix} \)
   \( C= \begin{bmatrix} 3&0&0\\0&3&0\\0&0&3\end{bmatrix} \)
   are scalar matrices of order 1, 2 and 3 respectively.
8. Unit Matrix
   If a square matrix has all elements 0 and each diagonal elements are 1, it is called identity matrix and denoted by I .
   For Example,
   \( A=\begin{bmatrix} 1\end{bmatrix} \)
   B=\( \begin{bmatrix} 1&0\\0&1\end{bmatrix}\)
   C= \( \begin{bmatrix} 1&0&0\\0&1&0\\0&0&1\end{bmatrix} \)
   are all zero matrices of the order 1, 2 and 3 respectively.
   Observe that a scalar matrix is an identity matrix when k = 1.
   Here, every identity matrix is clearly a scalar matrix, but the converse may not be true.
9. Zero Matrix
   A matrix is said to be zero matrix or null matrix if all its elements are zero.
   For Example,
   \( A=\begin{bmatrix} 0\end{bmatrix} \)
   B= \( \begin{bmatrix} 0&0\\0&0\end{bmatrix}\)
   C= \( \begin{bmatrix} 0&0&0\\0&0&0\\0&0&0\end{bmatrix}\)
   are all zero matrices of the order 1, 2 and 3 respectively.
10. Triangular Matrix
    There are two types of triangular Matrix.
    Upper Triangular Matrix:
    A square matrix in which all the elements below the diagonal are zero is known as the upper triangular matrix.
    For example,
    \( A=\begin{bmatrix} 3&0&0\\-5& 4& 0\\7& 0& 9\end{bmatrix} \)
    Lower Triangular Matrix:
    A square matrix in which all the elements above the diagonal are zero is known as the upper triangular matrix.
    For example,
    \( A=\begin{bmatrix} 3&0&-5\\0&4&7\\0&0&9\end{bmatrix}\)
11. Symmetric Matrix
    A matrix is symmetric only if it is square. If the transpose of a matrix is equal to the itself. More precisely, a matrix is said to be skew symmetric if A'=A. Also, for the matrix,\( a_{ji} = a_{ij}\) (for all the values of i and j.
    Some examples of symmetric matrices are:
    \( P=\begin{bmatrix} 5&1\\1&−3\end{bmatrix} \)
    \( Q=\begin{bmatrix}−5&12&57\\12&6&23\\57&23&78\end{bmatrix} \)
12. Skew-Symmetric Matrix
    A matrix is skew symmetric only if it is square. If the transpose of a matrix is equal to the negative of itself. More precisely, a matrix is said to be skew symmetric if A'=-A. Also, for the matrix,\( a_{ji} = -a_{ij}\) for all the values of i and j. The diagonal elements of a skew symmetric matrix are equal to zero.
    Some examples of skew symmetric matrices are:
    \( P=\begin{bmatrix} 0&8\\-8&0\end{bmatrix} \)
    \( Q=\begin{bmatrix}0&12&57\\-12&0&23\\-57&-23&0\end{bmatrix} \)

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Matrix Multiplication

We know that, dividing 3 by 7 is the same as multiplying 3 by \(\frac{1}{7}\), the inverse of 7. In symbols, 3 ÷ 7 = 3 × \(\frac{1}{7}\), or 3 × \(7^{-1}\) are same
In order to understandm division in Matrix, we need to discuss multiplicative inverse, \( A^{−1}\) of a matrix A.
However, multiplication of real numbers is commutative, \(\frac{3}{7}= 3 \times 7^{−1}= 7^{−1} \times 3\), in the matrices, multiplication is NOT commutative

Context 1

A school has following students in its grade.

	School
Grade 8	20
Grade 9	10
Grade 10	8

The school receive Rs4000, Rs 5000, and Rs 6000 fee from grade 8,9 and 10 respectively. Use matrix multiplication to find the total amount generated by students.

Context 2

A contractor builds three kinds of houses, models A,B,C with choice of two styles; structure(S) and prefab(P). Matrix A shows the number of each kind of houses planed for 100 new home subdivision.
A=\( \begin{matrix} \text{Model A} \\ \text{Model B}\\ \text{Model C} \end{matrix} \begin{bmatrix} \overset{\text{S}} {0} & \overset{\text{P}}{30}\\10&20\\20&20 \end{bmatrix}\)
The amount for each of the materials are shown in matrix B (Concrete, Cement, Brick, Decoration.
B=\( \begin{matrix} \text{S} \\ \text{P}\end{matrix} \begin{bmatrix} \overset{\text{Concrete}} {100} &\overset{\text{Cement}} {1000} &\overset{\text{Brick}} {40000} &\overset{\text{Decoration}} {2000} \\ 5&50&0&1000 \end{bmatrix}\)
Matrix C gives the cost for each kind of materials.
C=\(\begin{matrix} \text{Concrete}\\ \text{Cement}\\ \text{Brick}\\ \text{Decoration}\end{matrix} \begin{bmatrix} \overset{\text{Cost per Unit}} {10000} \\ 1000\\25\\25 \end{bmatrix}\)

Now, what is the contractor's total cost for materials required for various types of model homes?
Matrix Multiplication will be used to answer this question.

Multiplication of Matrices

In mathematics, matrix multiplication of A and B is defined only if the number of column in A is equal to the number of row in B.

Means, if A is an m x n matrix and B is an n x p matrix, then only the product AB is defined to be a new matrix AB with n x p order.

For example, if we have
\( A=\begin{bmatrix} a_{11}&a_{12}&a_{13} \\a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\\a_{41}&a_{42}&a_{43} \end{bmatrix}_{4 \times 3}\) and
\( B=\begin{bmatrix} b_{11}&b_{12}\\ b_{21}&b_{22}\\b_{31}&b_{32}\end{bmatrix}_{3 \times 2}\)
Then we get
\( AB=\begin{bmatrix} ab_{11}&{ab}_{12}\\{ab}_{21}&{ab}_{22}\\ {ab}_{31}&{ab}_{32}\\ {ab}_{41}&{ab}_{42} \end{bmatrix}_{4 \times 2} \)

Here, each i, j entry in AB is obtained by multiplying the entries
\( A_{ik}\) (row i of A) by the entries \( B_{kj}\) (column j of B), for k = 1, 2, ..., m , and summing the results.

For example,
\(ab_{11}\) entry in AB is obtained by
multiplying the entries \( A_{1k}\) ( row 1 of A) by the entries \( B_{k1}\) ( column 1 of B )

Key points of Matrix Multiplication

1. Multiplication of matrices is NON-commutative ; \( A \times B \ne B\times A\)
2. Multiplication of matrices is associative , \( A\times (B\times C) = (A\times B)\times C\) provided the multiplication is defined.
3. The order of matrix AB computed from A of order mxn and B of order nxp is mxp

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Example 1

Multiply the matrices \( A=\begin{bmatrix} 1&−1\\3&0\end{bmatrix}\) and \( B=\begin{bmatrix} 2&−1&1\\2&1&4\end{bmatrix}\)

Solution
Here, the multiplication of two matrix A and B is possible because the first matrix contains 2 columns and the second contains 2 rows.
Therefore, the product is
\( AB=\begin{bmatrix} 1 \times 2+(−1)\times2&1\times(−1)+(−1)\times1&1\times1+(−1)\times4\\ 3\times2+0\times2&3\times(−1)+0\times1&3\times1+0\times4\end{bmatrix} =\begin{bmatrix} 0&−2&-3\\6&-3&3\end{bmatrix} \)

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Determinant

The determinant of a square matrix is a real number or a function of the elements of the matrix which will represent area about the matrix. It determines also whether the system has a unique solution and whether the matrix is singular or not.
Thus, the determinant is a scalar value computed from the elements of a square matrix.

Let \(\vec{p}=(a,b)\) and \(\vec{q}=(c,d)\) are two vectors then the area of a parallelogram determined by the vectors \(\vec{p}\) and \(\vec{q}\) is represented by the matrix given by
\( M=\begin{pmatrix} a & c \\ b& d \end{pmatrix} \)
It is given by the formula
\( |M|=ad-bc \)

Drag the point p and q

The proof is given as
|M|= Area of parallelogram determined by \(\vec{p}\) and \(\vec{q}\)
or \(|M|= (a+c). (b+d) -\) area(red, blue and yellow regions)
or \(|M|= (ab+ad+cb+cd) -\) area(red)-area(blue)-area(yellow)
or \( |M|= (ab+ad+cb+cd) - 2bc-ab-cd \)
or |M|= ad-bc

Meaning of Determinant

Determinant एक स्केलर मान हो जसलाई सन्दर्भको आधारमा विभिन्न रूपमा व्याख्या गर्न सकिन्छ। यहाँ केहि ज्यामितीय व्याख्याहरू छन्:

1. Area and Volume:
   2x2 म्याट्रिक्सको determinant ले क्षेत्रफल जनाउदछ। यदि determinant positive छ भने, orientation सुरक्षित गर्दछ; यदि determinant negative भने, orientation लाई प्रतिबिम्बित गर्दछ।
   3x3 म्याट्रिक्सको लागि, determinant ले 3D स्पेसमा भोल्युमलाई प्रतिनिधित्व गर्दछ।
2. linear transformation:
   linear transformation को सन्दर्भमा, determinant ले scale दिन्छ जसद्वारा transformation हरू विस्तार वा संकुचित हुन्छ। 1 ले भोल्युममा कुनै परिवर्तन गर्दैन, 0 ले स्थानान्तरणलाई तल्लो-आयामी सबस्पेसमा vanish गर्छ भनेर संकेत गर्छ।
3. Inversion:
   The absolute value of the determinant of a square matrix is related to whether the matrix is invertible. If the determinant is non-zero, the matrix is invertible, and the absolute value of the determinant represents the volume scaling factor. If the determinant is zero, the matrix is singular, and the transformation collapses space.
4. Orientation: The sign of the determinant can indicate the orientation change caused by a linear transformation. Positive determinants preserve orientation, while negative determinants imply a reversal of orientation.

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Determinants of 2x2 and 3x3 matrix

The determinant of a matrix A is denoted det(A), det A, or |A|.
There are number of ways to compute determinants of matrices. Some of them are basketweave method, butterfly method, Sarrus’ method etc. The simplest among these methods is the basketweave method which could be stated as the combination of butterfly method for determinant computation of 2x2 matrices and Sarrus’ rule for determinant computation of 3x3 matrices.

A 2×2 matrix is written as
\( A=\begin{bmatrix} a_{11}& a_{12}\\ a_{21}& a_{22}\\ \end{bmatrix}\)

In order to find the determinant of the 2x2 matrix, we carry out the diagonal products, then subtract the non-diagonal product (right to left) from leading diagonal product (left to right) as as follows:
\( |A|=\begin{vmatrix} a_{11}& a_{12}\\a_{21}& a_{22}\\\end{vmatrix}\)
or \( |A|=a_{11} \times a_{22}- a_{21} \times a_{12}\)

Example 1

Find the determinant of a matrix \( A=\begin{bmatrix} 1 & -4 \\ 6 & 1 \\ \end{bmatrix} \)

Solution
The determinant of the matrix A is
\( det(A)=1 \times 1-(-4) \times 6=27 \)

NOTE

The determinant can be found based on the formal definition including the term minor, cofactor, adjoint.This is described as below.

Minors

In Matrix, minor of the entry in the i-th row and j-th column is the determinant of the submatrix formed by deleting the i-th row and j-th column.

This number is often denoted by \( M_{ij}\).

In the matrix below, the Minor of 1 is the determinant of the matrix

Cofactor

In Matrix, cofactor of the entry in the i-th row and j-th column is the product of
\( (-1)^{i+j} M_{ij}\)

Adjoint

Adjoint of matrix is the transpose of cofactor matrix

Determinant

Determinant of matrix A is denoted by det(A) or |A| and defined by
det(A)=Adjoint x Cofactor

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Inverse of Matrix

Inverse matrix is used to solve the system of linear equations. It is frequently used to encrypt message codes. Matrices are used by programmers to code or encrypt letters. A message is made up of a series of binary numbers that are solved using coding theory for communication and then an inverse matrix is used to decrypt the encoded message.

Inverse of Matrix

Three brands of fertilizer are available that provide nitrogen, phosphoric acid, and soluble potash to the soil. One bag of each brand provides the following units of each nutrient.
Nutrient \( \begin{pmatrix}&Brand A&Brand B&Brand C\\ Nitrogen&1&2&3\\ Phosphoric acid&3&1&2\\ Potash&2&0&1\end{pmatrix}\)
For ideal growth, the soil on a Farmer's farm needs 18 units of nitrogen, 23 units of phosphoric acid, and 13 units soluble potash per ropani.
Now, How many bags of each brand of fertilizer should be used per ropani for ideal growth of the Farmer's farm?

Inverse Matrix and its Multiplication will be used to answer this question.

For a square matrix A, the inverse is written \( A^{-1}\) When A is multiplied by \( A^{-1}\) the result is the identity matrix I. Non-square matrices do not have inverses.

Inverse of A is denoted by \( A^{-1}\) and defined by \( A^{-1}=\frac{1}{D} Adj(A)\) which satisfy:
\( A \times A^{-1} = A^{-1} \times A = I\)

Note:
Not all square matrices have inverses. A square matrix which has an inverse is called invertible or nonsingular, and a square matrix without an inverse is called non-invertible or singular matrix.

We can calculate the Inverse of a Matrix by:

1. Calculate Determinant
2. Calculating the Matrix of Minors
3. Calculating the Matrix of Cofactors
4. Calculating Adjoint
5. Find inverse

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Example

Find inverse of a matrix \( A=\begin{bmatrix} 4 & 3 \\ 3 & 2 \\ \end{bmatrix} \) 

Solution
Step 1: Calculate the Matrix of Minors
Minors is
Minor of \( \begin{bmatrix} 4 & 3 \\ 3 & 2 \\ \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 3 & 4 \\ \end{bmatrix} \)
For example, the minor of 4 = 2, is a determinant by canceling 1st row and 1st column from A

Step 2: Calculate the Matrix of Cofactors,
Cofactor of A is
cofactor of A=\(\begin{bmatrix} 2 & 3 \\ 3 & 4 \\ \end{bmatrix} =\begin{bmatrix} 2 & -3 \\ -3 & 4 \\ \end{bmatrix} \)
For example, the cofactor of 4 =\( (-1)^{1+1} . 2 =2\) , is a minor element.\( (-1)^{i+j}\)

Step 3: Calculate Adjoint
Adjoint of A is
Adj of \( \begin{bmatrix} 2 & -3 \\ -3 & 4 \\ \end{bmatrix} =\begin{bmatrix} 2 & -3 \\ -3 & 4 \\ \end{bmatrix} \)
For example, the adjoint is a new matrix obtained by interchanging row and column.

Step 4: Find Determinant.
The det A is
det A=\( \begin{bmatrix} 2 & -3 \\ -3 & 4 \\ \end{bmatrix} =-1\)
For example, the determinant is 2x4 - (-3)x(-3) = -1

Step 5: Find inverse.
The inverse of A is
\( A^{-1} =\frac{1}{D} \begin{bmatrix} 2 & -3 \\ -3 & 4 \\ \end{bmatrix} =\begin{bmatrix} -2 & 3 \\ 3 & -4 \\ \end{bmatrix} \)

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Properties of the inverse matrix

The following are the properties of the inverse matrix:

1. If A and B are square matrices with the order n and their product is an identity matrix, i.e., AB=I=BA, then \(B=A^{–1}\).
2. If a square matrix A has an inverse (non-singular), then the inverse matrix is unique.
3. A square matrix A has an inverse matrix if and only if the determinant is not zero, i.e., |A|≠0. Similarly, the matrix A is singular (has no inverse) if and only if its determinant is zero, i.e., |A|=0.
4. If a square matrix A has an inverse, then the determinant of an inverse matrix is the reciprocal of the matrix determinant. i.e., \( |A^{–1}|=\frac{1}{|A|}\).
5. If a square matrix A has an inverse, for a scalar k≠0 then the inverse of a scalar multiple is equal to the product of their inverse, i.e., \( (kA)^{–1}=\frac{1}{k}A^{–1}\).
6. If a square matrix A has an inverse, then the transpose of an inverse matrix is equal to the inverse of the transposed matrix. i.e., \( (A^{–1})T=(AT)^{–1} \).
7. If A and B are square non-singular matrices both with the order n then the inverse of their product is equal to the product of their inverse in reverse order, i.e., \((AB)^{–1}=(B)^{–1}(A)^{–1}\)
8. Let A and B are square matrices with the order n. If AB=O then either A=O or B=O or both A and B are singular matrices with no inverse.

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Cramer's rule

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.

It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand-sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750.

Given the system of equations
\( a_1 x+b_1 y=c_1 \)
\( a_2 x+b_2 y=c_2 \)
Solving, we can write
\( x=\frac{ \begin{vmatrix} c_{1} & b_{1} \\ c_{2} & b_{2} \\ \end{vmatrix} }{ \begin{vmatrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ \end{vmatrix} }\) and \( y=\frac{ \begin{vmatrix} a_{1} & c_{1} \\ a_{2} & c_{2} \\ \end{vmatrix} }{ \begin{vmatrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ \end{vmatrix} }\)
Suppose,
\( D=\begin{vmatrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ \end{vmatrix} \)
\( D_x =\begin{vmatrix} c_{1} & b_{1} \\ c_{2} & b_{2} \\ \end{vmatrix} \) and \( D_y = \begin{vmatrix} a_{1} & c_{1} \\ a_{2} & c_{2} \\ \end{vmatrix} \)
Then
\( x=\frac{D_x }{D},y=\frac{D_y }{D}\)

Solution

Given the system of equations
\( a_1 x+b_1 y=c_1 \)
\( a_2 x+b_2 y=c_2 \)
We can write
\( x \begin{bmatrix} a_1 \\ a_2 \\ \end{bmatrix} +y \begin{bmatrix} b_1 \\ b_2 \\ \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \\ \end{bmatrix} \)

The area of a parallelogram determined by \( \begin{bmatrix} a_1 \\ a_2 \\ \end{bmatrix} \) and \( \begin{bmatrix} b_1 \\ b_2 \\ \end{bmatrix} \) is given by
\( \begin{vmatrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ \end{vmatrix} \) (1)

The area of the parallelogram determined by \( x \times \begin{bmatrix} a_1 \\ a_2 \\ \end{bmatrix} \) and \( \begin{bmatrix} b_1 \\ b_2 \\ \end{bmatrix} \) is given by
\( \begin{vmatrix} xa_{1} & b_{1} \\ xa_{2} & b_{2} \\ \end{vmatrix} \) (2)

The area of the parallelogram determined by \( \begin{bmatrix} x a_1 \\ x a_2 \\ \end{bmatrix} + \begin{bmatrix} y b_1 \\ y b_2 \\ \end{bmatrix} \) and \( \begin{bmatrix} b_1 \\ b_2 \\ \end{bmatrix} \) is given by
\( \begin{vmatrix} xa_{1}+ y b_1& b_{1} \\ xa_{2} +y b_2 & b_{2} \\ \end{vmatrix} \) (3)

By Cavalieri's principle, the area of both parallelogram given by (2) and (3) is same.

Thus, equating the areas of this parallelograms, we get
\( \begin{vmatrix} xa_{1} & b_{1} \\ xa_{2} & b_{2} \\ \end{vmatrix} = \begin{vmatrix} xa_{1}+ y b_1& b_{1} \\ xa_{2} +y b_2 & b_{2} \\ \end{vmatrix} \)
or \( x \begin{vmatrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ \end{vmatrix} = \begin{vmatrix} c_1& b_{1} \\ c_2 & b_{2} \\ \end{vmatrix} \)
or \( x=\frac{ \begin{vmatrix} c_{1} & b_{1} \\ c_{2} & b_{2} \\ \end{vmatrix} }{ \begin{vmatrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ \end{vmatrix} }\)

Similarly, we can find
\( y=\frac{ \begin{vmatrix} c_{1} & a_{1} \\ c_{2} & a_{2} \\ \end{vmatrix} }{ \begin{vmatrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ \end{vmatrix} }\)
This is the Cramer's rule.

Example

A group of students took a trip on a bus, at Rs30 per child and Rs32 per adult for a total of Rs1184. They took the Taxi back at Rs35 per child and Rs36 per adult for a total of Rs1352. How many children, and how many adults?

Solution

This is just like
XA = B
or \( \begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix} {30} & {35} \\ {32} & {36} \\ \end{bmatrix} = \begin{bmatrix} 1184 & 1352 \end{bmatrix} \)
Now we have the inverse we can solve using:
\( X = BA^{-1} \)
or \( \begin{bmatrix} x & y \end{bmatrix} = \left [\begin{matrix} {30} & {35} \\ {32} & {36} \\ \end{matrix} \right ] ^{-1} \begin{bmatrix} 1184 & 1352 \end{bmatrix} \)