Maths Matrices and Determinants Chapter - Maths JEE Main

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.

JEE Main Maths Chapters 2024 

Important Topics of Matrices and Determinants Chapter

• Matrices

• Matrix Addition

• Square Matrix

• Matrix Operations

• Matrix Multiplication

• Elementary Operation of Matrix

• Properties of Determinant

• Determinant of a 3 x 3 Matrix

Matrices and Determinants Important Concepts for JEE Main

Definition of 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.

$A= \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$

Answer: 0

In the given matrix, each element in the ‘i’th row and ‘j’th column is represented by $a_{ij}$. This matrix is commonly represented as $[a_{ij}]_{m\times n}. The elements $a_{11}, a_{22}, a_{33}$, and so forth are referred to as the diagonal elements, and the summation of these diagonal elements is referred to as the trace of matrix A, which is symbolized as Tr(A).

Types of Matrices

• Column matrix – A matrix having only one column - \[\begin{bmatrix} 3 \\ 5 \\ 2 \\\end{bmatrix}\]

• Row matrix – A matrix having only one row - \[\begin{bmatrix} 1 & 5 & 3 \\ \end{bmatrix}\]

• Rectangular matrix – A matrix that has an unequal number of columns and rows -\[\begin{bmatrix} 6 & 8 \\ 0 & 1 \\ 3 & 2 \\ \end{bmatrix}\]

• Square matrix – A matrix having an equal number of columns and rows - \[\begin{bmatrix} 5 & 0 \\ 3 & 1 \\ \end{bmatrix}\]

• Diagonal matrix – A square matrix whose all elements except those in the main diagonal are zero -\[\begin{bmatrix} 2 & 0 \\ 0 & 3 \\ \end{bmatrix}\], \[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 3 \\ \end{bmatrix}\]

• Scalar matrix – A square matrix whose diagonal elements are all equal and all elements except those in the main diagonal are zero - \[\begin{bmatrix} 3 & 0 \\ 0 & 3 \\ \end{bmatrix}\], \[\begin{bmatrix} 7 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 7 \\ \end{bmatrix}\]

• Identity matrix – A square matrix whose main diagonal elements are ‘1’ and the other elements are all zero. It is denoted by ‘I’- \[\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}\], \[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\]

• Null matrix – A matrix of any order, all of whose elements are zero- \[\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix}\], \[\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}\]

• Triangular matrix – It is a square matrix in which all the elements above or below the main diagonal are zero - \[\begin{bmatrix} 1 & 0 & 0 \\ 7 & 1 & 0 \\ 3 & 4 & 3 \\ \end{bmatrix}\], \[\begin{bmatrix} 1 & 3 & 1 \\ 0 & 3 & 3 \\ 0 & 0 & 2 \\ \end{bmatrix}\]

• Transpose of a matrix – The matrix obtained by interchanging the rows and columns of a matrix A and is denoted by \[A^{T}\] −if A=\[\begin{bmatrix} 1 & 2 & 0 \\ 6 & 1 & 7 \\ 5 & 0 & 3 \\ \end{bmatrix}\] its transpose \[A^{T}\] = \[\begin{bmatrix} 1 & 6 & 5 \\ 2 & 1 & 0 \\ 0 & 7 & 3 \\ \end{bmatrix}\]

• Inverse of a Matrix – Inverse of a matrix usually applies for square matrices, and there exists an inverse matrix for every m×n square matrix. If the square matrix is represented by A, then its inverse is denoted by A-1, and it satisfies the property,

\[AA^{-1}\]= \[A^{-1}\]A = I, where I is the identity matrix.

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

Operations on Matrices

The following operations can be performed between two or more matrices:

• Addition of matrix 

• Subtraction of matrix

• Multiplication of matrix

• There is no division in matrices.

Elementary Operations of a Matrix

(i) Swapping Rows or Columns:

If you want to switch the places of two rows or two columns in a matrix, we write it as "Row i ↔ Row j" or "Column i ↔ Column j." It means you're moving the contents of row i to where row j was, and vice versa, or doing the same for columns.

(ii) Scaling a Row or Column:

If you want to make all the numbers in a row or column bigger or smaller by multiplying them with a number (except zero), you write it as "Row l → Row i (k)" or "Column l → Column i (k)." It means you're changing row l (or column l) by multiplying all its numbers by the same value k.

(iii) Adding a Multiple of One Row or Column to Another:

If you want to add some multiple of the numbers in one row or column to another row or column, you write it as "Row i → Row i + k * Row j" or "Column i → Column i + k * Column j." It means you're taking the numbers in row j (or column j), multiplying them by a value k, and then adding the result to row i (or column i).

Determinant of a Matrix

Determinant of a matrix is defined as a scalar value that can be calculated from the elements of a square matrix. It encodes some of the properties of the linear transformation that the matrix describes and is indicated as det A, det (A), or |A|. Let us clarify it further:

A square matrix A of a specific order has a number associated with it, and this number is called the determinant of matrix A. In order for a determinant to be associated with a matrix, the latter has to be a square matrix.

Thus, for the 2×2

square matrix A=\[\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \\ \end{bmatrix}\]

the symbol |A|=\[\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \\ \end{bmatrix}\]

signifies a determinant of second- order. 

Its value is defined as: \[\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \\ \end{bmatrix}\]= a1b2−a2b1

Similarly, for a 3 x 3 square matrix A=\[\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{bmatrix}\]

the symbol |A|=\[\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{bmatrix}\]

signifies a determinant of third-order. 

Its value is defined as:

|A|=a1\[\begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \\ \end{vmatrix}\] −b1\[\begin{vmatrix} a_2 & c_2 \\ b_3 & c_3 \\ \end{vmatrix}\] +c1\[\begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \\ \end{vmatrix}\]

Properties of Determinant

• If all the elements of a matrix are zero, then the determinant of the matrix is zero.

• If two rows or two columns are identical, then the determinant of the matrix is zero.

• For an identity matrix I of the order m×n, determinant of I, |I|= 1.

• If the matrix A has a transpose \[A^{T}\], then |\[A^{T}\]| = |A|.

• If $A^{-1}$ is the inverse of matrix A, then |\[A^{-1}\]| = 1/|A| = \[|A|^{-1}\]

• If two square matrices A and B are of the same size, then |AB| = |A| |B|.

• If c is a constant and a matrix A has the size b×b, then |cA| = \[c^{b}\] |A|.

• For triangular matrices, the product of the diagonal elements gives the determinant of the matrix.

• If | λA | = λn| A |, when A = [aij]n×n.

• If A and B are two square matrices of the same order, then | AB |= | A || B |

2 x 2 Matrix Determinant

Also commonly known as a determinant of a square matrix. A 2x2 matrix has two columns and two rows. The example mentioned above is an example of a 2x2 matrix determinant.

det(A) = \[\begin{vmatrix}a_{11} &b_{12} \\ c_{21} & d_{22} \end{vmatrix}\]

det(A) = a11 x a22  -  a12 x a21

3x3 Matrix Determinant

A 3x3 matrix determinant has three columns and three rows. The method involving solving for the 3x3 determinant matrix is different from what has been discussed until now in this article.

det (A) = \[\begin{vmatrix}a_{11} &a_{12}  &a_{13} \\a_{21} &a_{22}  &a_{23} \\a_{31}  &a_{32}  &a_{33} \end{vmatrix}\]

In order to solve for a 3x3 matrix determinant, follow the steps mentioned below:

Step 1: By expanding any one row, the solution for the determinant can be derived

Step 2: For solving det(A), the first row will be expanded

Step 3: The expanded version of the determinant will be as follow:

 a11 \[\begin{vmatrix}a_{22} &b_{23} \\ a_{32} & a_{33} \end{vmatrix}\]   -  a12\[\begin{vmatrix}a_{21} &b_{23} \\ a_{31} & a_{33} \end{vmatrix}\] + a13\[\begin{vmatrix}a_{21} &b_{22} \\ a_{31} & a_{32} \end{vmatrix}\]

Step 4: The 2x2 determinant matrix will be solved as mentioned above.

Step 5: After solving for those, simple multiplication with the items of row 1 can lead to the next and final step

Step 6: In the 3x3 determinant matrix, the signs are alternative ( +, -, +). Following these signs, one can get to the final answer of det(A). 

Cramer’s rule:

(i) Two variables:

• Consistent equations: Definite and unique solution. [Intersecting lines]

• Inconsistent equation: No solution. [Parallel line]

• Dependent equation: Infinite solutions. [Identical lines]

If, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}$

Then the given equations are inconsistent.

If,$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}= \frac{c_{1}}{c_{2}}$

Then the given equations are dependent.

(ii) Three variables:

Let $a_1x + b_1y + c_1 = 0; a_2x + b_2y + c_2 = 0$ and $a_3x + b_3y + c_3 = 0$, then $x = \dfrac{D_1}{D}, y = \dfrac{D_2}{D}$ and $z = \dfrac{D_3}{D}$, where 

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

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

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

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

(iii) Consistency of a system of equations:

(a) If the number D is not zero, and at least one of D1, D2, or D3 will not be zero, then the set of equations you're looking at has a solution that's unique and non-trivial solution.

(b) If the number D is not zero, but all three D1, D2, and D3 are equal to zero, then the equations are consistent and has trivial solution only.

(c) If all D, D1, D2, and D3 are equal to zero, then the equations can either have infinite solutions or no solution.

(d) If D is zero, but at least one of D1, D2, or D3 is not zero, then the equations don't have a solution. They just can't be solved consistently.

(e) When a set of linear equations has only the solution where all variables are equal to zero, we call that the "trivial solution."

(iv) Three equation in two variable:

If x and y are not equal to zero, then the way to make sure that the equations $a_1x + b_1y + c_1 = 0; a_2x + b_2y + c_2 = 0$ and $a_3x + b_3y + c_3 = 0$ to be consistent in x and y is

$\begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} &c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} = 0$

JEE Main Matrices and Determinants Solved Example:

1. If A=\[\begin{bmatrix} 2 & 5 & -1 \\ 4 & 1 & 3 \\ \end{bmatrix}\] and B =\[\begin{bmatrix} 3 & 2 & 4 \\ -1 & 5 & 2 \\ \end{bmatrix}\], 

then A + B  = \[\begin{bmatrix} 2 & 5 & -1 \\ -1 & 5 & 2 \\ \end{bmatrix}\] +\[\begin{bmatrix} 3 & 2 & 4 \\ -1 & 5 & 2 \\ \end{bmatrix}\] 

=\[\begin{bmatrix} 2+3 & 5+2 & -1+4 \\ 4-1 & 1+5 & 3+2 \\ \end{bmatrix}\] 

=\[\begin{bmatrix} 5 & 7 & 3 \\ 3 & 6 & 5 \\ \end{bmatrix}\] 

2. If A = \[\begin{bmatrix} 1 & 3 & -2 \\ 4 & 7 & 5 \\ \end{bmatrix}\]

and B =\[\begin{bmatrix} -2 & 1 & -1 \\ 3 & 5 & 2 \\ \end{bmatrix}\] 

then A-B = A+(-B) =\[\begin{bmatrix} 1 & 3 & -2 \\ 4 & 7 & 5 \\ \end{bmatrix}\] + \[\begin{bmatrix} 2 & -1 & 1 \\ -3 & -5 & -2 \\ \end{bmatrix}\]

= \[\begin{bmatrix} 1+2 & 3-1 & -2+1 \\ 4-3 & 7-5 & 5-2 \\ \end{bmatrix}\]

= \[\begin{bmatrix} 3 & 2 & -1 \\ 1 & 2 & 3 \\ \end{bmatrix}\]

3. If A=\[\begin{bmatrix} 1 & 1 \\ 0 & 2 \\ 1 & 1 \\ \end{bmatrix}\] and B  = \[\begin{bmatrix} 1 & 2 \\ 2 & 2 \\ \end{bmatrix}\], 

Then 

AB = \[\begin{bmatrix} 1 & 1 \\ 0 & 2 \\ 1 & 1 \\ \end{bmatrix}\] \[\begin{bmatrix} 1 & 2 \\ 2 & 2 \\ \end{bmatrix}\]

=\[\begin{bmatrix} 1\times 1+1\times 2 & 1\times 2+1\times 2 \\ 0\times 1+2\times 2 & 0\times 2+2\times 2 \\ 1\times 1+1\times 2 & 1\times 2+1\times 2 \\ \end{bmatrix}\] \[\begin{bmatrix} 3 & 4 \\ 4 & 4 \\ 3 & 4 \\ \end{bmatrix}\]

4. Find the determinant of matrix A, if A=\[\begin{bmatrix} 2 & 5 \\ 1 & 3 \\ \end{bmatrix}\] 

Solution: |A|= \[\begin{bmatrix} 2 & 5 \\ 1 & 3 \\ \end{bmatrix}\] 

(2 x 3) - (5 x 1) = 6 - 5 = 1

5. Solve for det(A) which is \[\begin{vmatrix}5 &2  &1 \\-2 &-1  &1 \\-4  &4  &3 \end{vmatrix}\]

For solving det(A), row 1 shall be expanded

That being said, the expanded version of this determinant is given below

5\[\begin{vmatrix}-1 &1 \\ 4 & 3 \end{vmatrix}\]   -2\[\begin{vmatrix}-2 &1 \\ -4 & 3 \end{vmatrix}\] + 1\[\begin{vmatrix}-2 &-1 \\ -4 & 4 \end{vmatrix}\]

This step involves solving for the 2x2 determinant matrices

5 { (-1 x 3) - (4 x 1)}  - 2 { ( -2 x 3 ) - ( 1 x -4)}  + 3 {(-2 x 4) - ( -1 x -4)}

5 ( -3 - 4) - 2 (-6 + 4) + 3( -6 -4)

5(-7) -2(-2) +3(-10)

= -35 + 4 -30 = -61

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Conclusion

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.