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Matrix and Determinant is one of the most intriguing, simple, and significant subjects in mathematics. Every year, at least 1 - 3 problems from this chapter will appear in JEE Main and other entrance examinations. This chapter is completely new from the student's perspective, as it will be covered in 12th grade.

Determinants and matrices are used to solve linear equations by applying Cramer's rule to a collection of non-homogeneous linear equations. Only square matrices are used to calculate determinants. When a matrix's determinant is zero, it's known as a singular determinant, and when it's one, it's known as unimodular. The determinant of the matrix must be nonsingular (i.e., its value must be non-zero) for the system of equations to have a unique solution. Let us look at the definitions of determinants and matrices, as well as the various types of matrices and their properties, using examples.

Matrices

Matrix Addition

Square Matrix

Matrix Operations

Matrix Multiplication

Elementary Operation of Matrix

Properties of Determinant

Determinant of a 3 x 3 Matrix

Matrices are a type of ordered rectangular array of numbers used to represent linear equations. There are rows and columns in a matrix. On matrices, we can execute mathematical operations such as addition, subtraction, and multiplication. The number of rows and columns in a matrix represents the order of the matrix. Let there are m rows and n columns in a matrix, then its order will be m x n.

$A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \cdots & a_{1 n} \\ a_{21} & a_{22} & a_{23} \cdots & a_{2 n} \\ a_{31} & a_{32} & a_{33} \cdots & a_{3 n} \\ : & : & : & : \\ a_{m 1} & a_{m 2} & a_{m 3} \cdots & a_{m n} \end{bmatrix}$

Column Matrix – A matrix having only one column.

Example: $\begin{bmatrix} 3 \\ 5 \\ 2 \end{bmatrix}$

Row Matrix – A matrix having only one row.

Example: $\begin{bmatrix} 3 & 5 & 2 \end{bmatrix}$

Rectangular Matrix – A matrix that has an unequal number of columns and rows.

Example: $\begin{bmatrix} 6 & 8 \\ 0 & 1 \\ 3 & 2 \end{bmatrix}$

Square Matrix – A matrix having an equal number of columns and rows.

Example: $\begin{bmatrix} 5 & 0 \\ 3 & 1 \end{bmatrix}$

Null/Zero Matrix – Matrix that has all its elements equal to zero.

Example: $\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$

Identity Matrix – A square matrix with 1’s on the main diagonal and other elements are zero.

Example: $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Scalar Matrix – A square matrix whose one diagonal (main diagonal) elements are equal and all elements except those in the main diagonal are zero. An identity matrix is also scalar matrix.

Example: $\begin{bmatrix} 7 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 7 \end{bmatrix}$

Diagonal Matrix – A square matrix whose all elements except those in the main diagonal are zero.

Example: $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 3 \end{bmatrix}$

Upper Triangular Matrix – A square matrix in which all entries above the main diagonal are 0.

Example: $\begin{bmatrix} 2 & -6 & 3 \\ 0 & 4 & 9 \\ 0 & 0 & 7 \end{bmatrix}$

Lower Triangular Matrix – A square matrix in which all entries below the main diagonal are 0.

Example: $\begin{bmatrix} 2 & 0 & 0 \\ 4 & 9 & 0 \\ 3 & -6 & 7 \end{bmatrix}$

Inverse of a Matrix – The inverse of a matrix mainly applies to square matrices, and every m x n square matrix has an inverse matrix. If A denotes the square matrix, then $A^{-1}$ denotes its inverse, and it satisfies the property.

$A A^{-1}=A^{-1}{A}={I}$, where I is the identity matrix

Note: The determinant of the square matrix should be non-zero.

Addition, subtraction and multiplication are algebraic operations on matrices; there is no division in matrices. In addition, the Transpose and Conjugate of the matrix are two of the most essential operations that will be included in determinants.

Transpose of the Matrix: If A is a matrix, the transpose of the matrix is the matrix formed by replacing the columns of a matrix with rows or rows with columns.

Example: If $A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$, Then transpose of the matrix A will be, ${A}^{\prime}=\begin{bmatrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32} \\ a_{13} & a_{23} & a_{33} \end{bmatrix}$

Conjugate of the Matrix: If an element of a matrix A is a complex number, the matrix formed by replacing that complex number with its conjugate is known as the conjugate of the matrix A and is represented by ${\overline A}$.

Example: If $A=\begin{bmatrix}i & 2+3 i & 4 \\3 i & 6 & 4+5 i \\4+5 i & 4 i & 2+3 i\end{bmatrix}$, Then conjugate of the matrix A will be, $\overline{A}=\begin{bmatrix}- i & 2-3 i & 4 \\-3 i & 6 & 4-5 i \\4-5 i & -4 i & 2-3 i\end{bmatrix}$

Determinants take a square matrix as input and produce a single number as output.

For any square matrix, M = $\left[a_{ij}\right]$ of order n x n, a determinant can be defined as a real scalar value or a complex number, where $a_{ij}$ is the (i, j)th element of the matrix C.

Method to Solve Determinant by Considering the Top Row Items and Their Minors

Consider 3 x 3 square matrix

$A=\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ a_4 & b_4 & c_4 \end{bmatrix}$

Determinant represents, $|A|$ or $\text{det}(A)=\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$

The anchor number $a_1$ is fixed, as is the 3 x 3 determinant of its sub-matrix (minor of $a_1$). Calculate the minors of $b_1$ and $c_1$ in the same way.

Multiply the small determinant by the anchor number and the sign of the anchor number.

$\begin{vmatrix} + & - & + \\ - & + & - \\ + & - & + \end{vmatrix}$

Plus $a_1$ times the 2 x 2 matrix's determinant obtained by deleting the row and column containing $a_1$.

Minus $b_1$ times the 2 x 2 matrix determinant obtained by deleting the row and column containing $b_1$

Plus $c_1$ times the 2 x 2 matrix's determinant obtained by deleting the row and column containing $c_1$

Minus $d_1$ times the determinant of the 2 x 2 matrix obtained by deleting the row and column containing $d_1$

We finally get,

$|A|=a_{1} \cdot\begin{vmatrix} b_{2} & c_{2} \\ b_{3} & c_{3} \end{vmatrix} - b_{1} \cdot \begin{vmatrix} a_{2} & c_{2} \\ a_{3} & c_{3} \end{vmatrix} + c_{1} \cdot\begin{vmatrix}a_{2} & b_{2} \\ a_{3} & b_{3} \end{vmatrix}$

$\Rightarrow |A|=a_{1}\left(b_{2} c_{3}-b_{3} c_{2}\right)-b_{1}\left(a_{2} c_{3}-a_{3} c_{2}\right)+c_{1}\left(a_{2} b_{3}-a_{3} b_{2}\right)$

Trick Used: Multiply the first element in the top row by its minor, then subtract the product of the second element and its minor from the result. Continue to add and subtract the product of each element of the top row with its respective minor until all of the top row's constituents have been taken into account.

Elementary transformation of matrices is very important. It's used to find equivalent matrices and the inverse of a matrix, among other things. Playing with the rows and columns of a matrix is an elementary transformation.

Elementary Row Transformation

Only the rows of the matrices are transformed, with no changes to the columns. These row operations are carried out in accordance with a set of rules that ensure the transformed matrix is equivalent to the original matrix.

Without using any formula like $A^{-1} = \dfrac{adj A}{det A}$, the elementary row transformations are also used to find the inverse of a matrix A.

We usually refer to the first row as $R_1$, the second row as $R_2$, and so on when performing elementary row operations. There are three different types of basic row operations:

1. Interchanging two rows.

$R_1 \leftrightarrow R_2$ represents swapping the first and second rows.

2. Multiplying/dividing a row by a scalar.

Suppose if the first row is multiplied with all the rows by a scalar, say 2, then it is represented as $R_1 \rightarrow 2R$.

3. Adding/subtracting to the corresponding elements of another row after multiplying/dividing a row by some scalar.

Suppose the first row is multiplied by 4, and then added to the second row, we represent it as $R_1 \rightarrow 4R_1+R_2$ or $R_2 \rightarrow R_2+3R_1$.

Elementary Column Operation

Because we've already gone over row transformation in depth, we'll only go over column transformation briefly. The basic column transformation rules are as follows:

1. When any two columns are interchanged, $C_1 \leftrightarrow C_2$

2. Any non-zero number can be multiplied by all the elements of any column, $C_1 \rightarrow 3C$

3. All the column's elements can be multiplied by any non-zero constant and added to corresponding elements of another column, $R_1 \rightarrow R_1+3R_2$

Property 1: If $I_n$ is the identity matrix of the order n x n, then det(I) = 1.

$I = \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} = 1(1) - 0 = 1$

Property 2: If $A^{T}$ is a transpose of a matrix A, then det($A^{T}$) = det(A).

$A = \begin{vmatrix} 1 & 3 \\ 5 & 7 \end{vmatrix} = 1(7) - 3(5) = -8$

$A^{T} = \begin{vmatrix} 1 & 5 \\ 3 & 7 \end{vmatrix} = 1(7) - 3(5) = -8$

Property 3: If there is a zero row or column in any square matrix B of order n x n, then det(B) = 0.

$B = \begin{vmatrix} 2 & 5 \\ 0 & 0 \end{vmatrix} = 2(0) - 5(0) = 0$

Property 4: If M is an upper-triangular matrix or a lower-triangular matrix, det(M) is the product of all of its diagonal entries.

$M = \begin{vmatrix} 3 & 6 & 9 \\ 0 & 2 & 4 \\ 0 & 0 & 1\end{vmatrix} = 3 \times 2 \times 1 = 6$.

Property 5: If the column or row of two square matrices is expressed as a sum of terms, it can be expressed as a sum of two determinants.

$\begin{vmatrix} a_{1}+b_{1} & a_{2}+b_{2} & a_{3}+b_{3} \\ c_{1} & c_{2} & c_{3} \\ d_{1} & d_{2} & d_{3} \end{vmatrix} = \begin{vmatrix} a_{1} & a_{2} & a_{3} \\ c_{1} & c_{2} & c_{3} \\ d_{1} & d_{2} & d_{3} \end{vmatrix} = \begin{vmatrix} b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \\ d_{1} & d_{2} & d_{3} \end{vmatrix}$

Property 6: If the size of matrix M is n x n and k is a constant then each element of a given row or column is then multiplied by a constant i.e., Det(kM) = k x Det(M)

$\begin{vmatrix} ka_{1} & kb_{1} & kc_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} = k \times \begin{vmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix}$

Property 7: The Adjugate Matrix (Adjoint of a Matrix) and Laplace's Formula

Adjugate Matrix: It is determined by transposing the matrix that contains the cofactors and is calculated using the equation:

$(Adj (M))_{x,y} = (-1)^{x+y} N_{x,y}$

Laplace’s Formula: Using this formula, the determinant of a matrix is expressed in terms of its minors. If the matrix $N_{x y}$ is the minor of matrix M, obtained by eliminating the xth and yth column and has a size of ( j-1 × j-1), then the determinant of the matrix M will be given as:

$|M| = \sum_{y=1}^{i} (−1)^{x+y} a_{x,y} N_{x,y}$ where $(−1)^{x+y} N_{x,y}$ is the cofactor.

A scalar value that may be computed from the elements of a square matrix is known as the determinant of a matrix. It is denoted as det A, det (A), or |A| and encodes some of the properties of the linear transformation that the matrix describes.

Determinant of 2 x 2 Matrix

Consider a matrix $A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$

Then, determinant is $|A|=\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$

Therefore $|A|=a_{11} a_{22}-a_{21} a_{12}$

Determinant of 3 x 3 Matrix

Consider a matrix $A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$

Then, determinant is $|A|=\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$

Therefore $|A| =a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} -a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} +a_{13}\begin{vmatrix}a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}$

One of the applications of Matrix and Determinant is the solution of linear equations in two or three variables. Matrices and determinants are also used to determine whether or not a system is consistent.

Example 1: Find the multiplication AB of a matrix A and B where $A = \begin{bmatrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{bmatrix} $ and $B = \begin{bmatrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{bmatrix} $

Solution: Given $A = \begin{bmatrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{bmatrix} $ and $B = \begin{bmatrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{bmatrix} $

To find AB multiply both the matrices and add the product of the terms

$AB = \begin{bmatrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 & 2 \\3 & 2 & 1 \\2 & 1 & 3 \end{bmatrix}$

$AB = \begin{bmatrix} (-5+3+6) & (-5+2+3) & (-10+1+9) \\(7+3-10) & (7+2-5) & (14+1-15) \\ (1-3+2) & (1-2+1) & (2-1+3) \end{bmatrix}$

$\therefore AB =\begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4\end{bmatrix}$

Example 2: Find the determinant of a matrix $\begin{vmatrix} 2 & -3 \\ 4 & 5 \end{vmatrix} $

Solution: Let us assume the determinant of matrix be $|A|$

So, $|A| = \begin{vmatrix} 2 & -3 \\ 4 & 5 \end{vmatrix}$

W.K.T $\text{det }A = a_{11} a_{22}-a_{21} a_{12}$

On substituting the values, we get

$|A|= 2 \times 5 - (-3) \times 4$

$\Rightarrow 10 + 12 = 22$

Example 3: Find the determinant of a matrix $M = \begin{vmatrix} 1 & 3 & 2 \\ -3 & -1 & -3 \\ 2 & 3 & 1 \end{vmatrix} $

Solution: Given, $|M| = \begin{vmatrix} 2 & -3 \\ 4 & 5 \end{vmatrix}$

As it is a 3 x 3 matrix W.K.T the determinant will be $|M|=a_{1} \cdot\begin{vmatrix} b_{2} & c_{2} \\ b_{3} & c_{3} \end{vmatrix} - b_{1} \cdot \begin{vmatrix} a_{2} & c_{2} \\ a_{3} & c_{3} \end{vmatrix} + c_{1} \cdot\begin{vmatrix}a_{2} & b_{2} \\ a_{3} & b_{3} \end{vmatrix}$

$|M|=a_{1}\left(b_{2} c_{3}-b_{3} c_{2}\right)-b_{1}\left(a_{2} c_{3}-a_{3} c_{2}\right)+c_{1}\left(a_{2} b_{3}-a_{3} b_{2}\right)$

On substituting the values, we get

$|M|= 1 \cdot \begin{vmatrix} -1 & -3 \\ 3 & 1 \end{vmatrix} - 3 \cdot \begin{vmatrix} -3 & -3 \\ 2 & 1 \end{vmatrix} + 2 \cdot \begin{vmatrix} -3 & -1 \\ 2 & 3 \end{vmatrix}$

Using determinants rule we get,

$|M|=1 \cdot(-1-(-9)-3 \cdot(-3-(-6)+2 \cdot(-9-(-2)) \\ \Rightarrow 1 \cdot(-1+9)-3 \cdot(-3+6)+2 \cdot(-9+2) \\ \Rightarrow 8-9-14 \\ \therefore |M|=-15$

1. If $A = \begin{bmatrix}a & b \\ b & a\end{bmatrix}$ the find $A^2$

Ans: Given $A = \begin{bmatrix}a & b \\ b & a\end{bmatrix}$

Then, $A^2 = \begin{bmatrix} a & b \\ b & a \end{bmatrix} \begin{bmatrix} a & b \\ b & a \end{bmatrix}$

$\Rightarrow \begin{bmatrix} a^{2}+b^{2} & 2 a b \\ 2 a b & a^{2}+b^{2} \end{bmatrix}$

$\therefore \alpha=a^{2}+b^{2}, \beta=2ab$

2. Find ${c}^{2}+{x}^{2}+{y}^{2}$ if the matrix A given by ${A}= \begin{bmatrix}{a} & \dfrac{2}{3} & \dfrac{2}{3} \\ \dfrac{2}{3} & \dfrac{1}{3} & {b} \\ {c} & {x} & {y}\end{bmatrix}$ is orthogonal.

Ans: Given matrix $A$ is orthogonal.

Therefore, $\begin{bmatrix}{a} & \dfrac{2}{3} & \dfrac{2}{3} \\ \dfrac{2}{3} & \dfrac{1}{3} & {b} \\ {c} & {x} & {y}\end{bmatrix}$

$\begin{bmatrix} {a} & \dfrac{2}{3} & {c} \\ \dfrac{2}{3} & \dfrac{1}{3} & x \\ \dfrac{2}{3} & b & y \end{bmatrix} $

As we can see that the diagonal matrix are same, so the given matrix can be written as,

$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

3. The system of equations $\alpha x+y+z=\alpha-1, x+\alpha y+z=\alpha-1 $ and $x+y+\alpha z=\alpha-1$ has no solution, if $\alpha$ is

Ans: The given equations can be written in the determinant form,

$A=\begin{vmatrix} \alpha & 1 & 1 \\ 1 & \alpha & 1 \\ 1 & 1 & \alpha\end{vmatrix}=0$

On solving the determinant we get,

$\alpha\left(\alpha^{2}-1\right)-1(\alpha-1)+1(1-\alpha)=0$

$\Rightarrow \alpha(\alpha-1)(\alpha+1)-1(\alpha-1)-1(\alpha-1)=0$

$\Rightarrow(\alpha-1)\left[\alpha^{2}+\alpha-1-1\right]=0$

$\Rightarrow(\alpha-1)\left[\alpha^{2}+\alpha-2\right]=0$

$\Rightarrow \left[\alpha^{2}+2 \alpha-\alpha-2\right]=0$

$\Rightarrow(\alpha-1)[\alpha(\alpha+2)-1(\alpha+2)]=0$

$\Rightarrow(\alpha-1)=0, \alpha+2=0$

$\Rightarrow \alpha=-2,1: \text { but } \alpha \neq 1$

1. Find the multiplication of the two matrices $A = \begin{bmatrix} 1 & 1 \\ 0 & 2 \\ 1 & 1 \end{bmatrix} $ and $B = \begin{bmatrix} 1 & 2 \\ 2 & 2 \end{bmatrix} $

Answer: $\begin{bmatrix} 3 & 4 \\ 4 & 4 \\ 3 & 4 \end{bmatrix}$

Trick to Solve: Take the elements from the first row of the first matrix and the elements from the first column of the second matrix. Multiply the corresponding elements and add all the products.

2. Find the determinant of $\begin{vmatrix}1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & -2 & 5 \end{vmatrix}$

Answer: 0

In this article, we have elaborated on concepts and solutions to questions on the topic Determinants and Matrices. We have also learn how to find determinant of a matrix (determinant of a matrix formula) and properties of matrices and determinants. Everything you're looking for is available in a single location. Students can carefully read through the concepts, definitions and questions in the PDFs, which are also free to download and understand the concepts used to solve these questions. This will be extremely beneficial to the students in their exams.

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FAQ

**1. What is the order of matrices and determinants?**

A matrix has an order of m x n because it has m rows and n columns, whereas a determinant has an order of n x n because it has n rows and n columns( it should have an equal number of rows and columns).

**2. Who is the father of matrices?**

Arthur Cayley, known as the "Father of Matrices," was a brilliant mathematician. On August 16, 1821, he was born. In 1858, Arthur Cayley presented the conceptual explanation of the matrix in his Memoir on the Theory of Matrices. As a result, matrices became one of the most important branches of mathematics in the research. He primarily worked on Algebra and was instrumental in the establishment of the modern British pure mathematics school.

**3. What are the unique uses of matrices and determinants?**

The matrix has numerous applications in data science and artificial intelligence. The matrix inversion method can be used to solve a large number of algebraic equations. A matrix's transpose, adjoint, and inverse can also be found.

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