Adjoint and Inverse of a Matrix: CBSE Class 12 Maths Determinants Ex 4.5

Key Definitions: Adjoint, Singular, and Non-Singular Matrices

Adjoint of a Matrix (adj A)

The adjoint of a square matrix A is the transpose of its cofactor matrix. If C is the matrix of cofactors of A, then adj(A) = Cᵀ. Each element (i, j) of the adjoint matrix is the cofactor of the element (j, i) of the original matrix A.

Singular Matrix

A square matrix A is called singular if its determinant is zero, i.e., |A| = 0. Singular matrices do not have an inverse.

Non-Singular Matrix

A square matrix A is called non-singular if its determinant is non-zero, i.e., |A| ≠ 0. Only non-singular matrices are invertible.

Inverse of a Matrix (A⁻¹)

For a non-singular square matrix A, its inverse, denoted by A⁻¹, is given by the formula A⁻¹ = (1/|A|) * adj(A). The inverse has the property that AA⁻¹ = A⁻¹A = I, where I is the identity matrix.

Understanding the Core Theorem: A(adj A) = |A|I

A fundamental theorem connects a matrix, its adjoint, and its determinant. For any square matrix A of order n, we have:

A(adj A) = (adj A)A = |A|I

Where I is the identity matrix of order n. But why is this true? Let's understand the intuition. When you multiply matrix A with its adjoint, the diagonal elements of the resulting matrix are the sum of the products of elements of a row (or column) with their corresponding cofactors. We know this sum is precisely the value of the determinant, |A|. The off-diagonal elements are the sum of the products of elements of a row (or column) with the cofactors of another row (or column). A key property of determinants states that this sum is always zero. Therefore, the resulting matrix is a diagonal matrix with |A| on the diagonal and zeros everywhere else, which is simply |A|I. This very theorem gives us the formula for the inverse. If |A| ≠ 0, we can divide by it: A * ( (1/|A|) adj A ) = I. This shows that the inverse, A⁻¹, must be (1/|A|) adj A.

Step-by-Step Process: How to Find the Inverse of a 3x3 Matrix

1. Step 1: Calculate the Determinant (|A|) — First, find the determinant of the matrix A. If |A| = 0, the matrix is singular, and the inverse does not exist. You can stop here. If |A| ≠ 0, proceed to the next step.
2. Step 2: Find the Cofactor Matrix (C) — Calculate the cofactor Cᵢⱼ for each element aᵢⱼ of the matrix A. Remember the formula Cᵢⱼ = (-1)ⁱ⁺ʲ Mᵢⱼ, where Mᵢⱼ is the minor of the element aᵢⱼ. This involves calculating 9 cofactors for a 3x3 matrix. Arrange them into a new matrix, the Cofactor Matrix C.
3. Step 3: Find the Adjoint Matrix (adj A) — The adjoint of A is the transpose of the cofactor matrix C. Simply interchange the rows and columns of C to get adj(A). So, adj(A) = Cᵀ.
4. Step 4: Calculate the Inverse (A⁻¹) — Use the final formula by dividing the adjoint matrix by the determinant: A⁻¹ = (1/|A|) * adj(A). This means each element of the adjoint matrix gets divided by the determinant |A|.

Exam Traps and Common Mistakes

Be very careful during calculations as this topic is prone to simple errors that can cost you marks.

• Forgetting to Check the Determinant: Always calculate |A| first. If you proceed to find the inverse of a singular matrix (|A|=0), you will lose marks.
• Sign Errors in Cofactors: The (-1)ⁱ⁺ʲ part is a common source of mistakes. A good practice is to remember the sign pattern matrix:

[[+, -, +], [-, +, -], [+, -, +]]

• Confusing Adjoint and Cofactor Matrix: A very frequent error is to forget to take the transpose of the cofactor matrix to get the adjoint. Always remember: adj(A) = (Cofactor Matrix)ᵀ.
• Forgetting the Final Division: After finding the adjoint, many students forget the last step of dividing by the determinant |A|. The inverse is (1/|A|) * adj(A), not just adj(A).

Practice Questions with Solutions

• Q: Find the adjoint of the matrix A = [[2, 3], [4, 1]]. A: Step 1: For a 2x2 matrix [[a, b], [c, d]], the adjoint is given by adj(A) = [[d, -b], [-c, a]]. Step 2: Here, a=2, b=3, c=4, d=1. Substitute these values into the formula. Final answer: adj(A) = [[1, -3], [-4, 2]]
• Q: Find the inverse of the matrix A = [[1, 0, 0], [0, 2, 0], [0, 0, 3]] using the adjoint method. A: Step 1: Calculate the determinant of A. |A| = 1(23 - 00) - 0 + 0 = 6. Step 2: Find the cofactor matrix. C11=6, C12=0, C13=0, C21=0, C22=3, C23=0, C31=0, C32=0, C33=2. Step 3: Find the adjoint of A, which is the transpose of the cofactor matrix. adj(A) = [[6, 0, 0], [0, 3, 0], [0, 0, 2]]. Step 4: Calculate A⁻¹ = (1/|A|)adj(A) = (1/6)[[6, 0, 0], [0, 3, 0], [0, 0, 2]]. Final answer: A⁻¹ = [[1, 0, 0], [0, 1/2, 0], [0, 0, 1/3]]
• Q: For the matrix A = [[2, 3], [1, 2]], verify that A(adj A) = |A|I. A: Step 1: Calculate the determinant of A. |A| = (22) - (31) = 4 - 3 = 1. So, |A|I = 1 [[1, 0], [0, 1]] = [[1, 0], [0, 1]]. Step 2: Find the adjoint of A. For a 2x2 matrix [[a, b], [c, d]], adj(A) = [[d, -b], [-c, a]]. Thus, adj(A) = [[2, -3], [-1, 2]]. Step 3: Calculate A(adj A) = [[2, 3], [1, 2]] [[2, -3], [-1, 2]] = [[(22)+(3-1), (2-3)+(32)], [(12)+(2-1), (1-3)+(22)]] = [[4-3, -6+6], [2-2, -3+4]] = [[1, 0], [0, 1]]. Final answer: Since A(adj A) = [[1, 0], [0, 1]] and |A|I = [[1, 0], [0, 1]], the identity is verified.
• Q: Solve the following system of linear equations using the matrix method: 2x + 5y = 1, 3x + 2y = 7. A: Step 1: Write the system in matrix form AX=B: A=[[2, 5], [3, 2]], X=[[x], [y]], B=[[1], [7]]. Step 2: Calculate |A| = (22) - (53) = 4 - 15 = -11. Since |A| != 0, A⁻¹ exists. Step 3: Find adj(A). For a 2x2 matrix, adj(A) = [[2, -5], [-3, 2]]. Step 4: Calculate A⁻¹ = (1/|A|)adj(A) = (-1/11)[[2, -5], [-3, 2]]. Step 5: Solve for X = A⁻¹B = (-1/11)[[2, -5], [-3, 2]] [[1], [7]] = (-1/11)[[21 + (-5)7], [(-3)1 + 2*7]] = (-1/11)[[2-35], [-3+14]] = (-1/11)[[-33], [11]] = [[3], [-1]]. Final answer: x = 3, y = -1.

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