Test: Adjoint and Inverse (May 5) - JEE MCQ

Determinant of (A^-1) = 
Where, Determinant of A = 1(4 × 1 - 5 × 0) - 2(0 - 0) + 3(0 - 0) = 4
Given:
Determinant of A = 4 
So, Determinant of (A^-1) = 

adj.A = Transverse of matrix 
A₁₁ = A_ij = (-1)ⁿ × (cross multiplication of remaining column and row) = (-1)ⁿ × 
Where, n = (i + j)
We follow the same procedure for the A₁₁, A₁₂, A₁₃, .......as so on. Also find the determinant of matrix A. Then applies the given formula after that we finally obtain the A^-1.

Apply element-wise inversion i.e multiply by the inverse of the same element to make it an identity matrix until the desired objective is reached.
Let (BA)^-1 = K
Multiply by the BA on both sides, in the same order
(BA)(BA)^-1 = (BA)K
I = BAK   {∵ (A)(A^-1) = I}
Multiply B^-1 on both the sides
B^-1I = B^-1BAK
B^-1 = AK
Multiply A^-1 on both sides
A^-1B^-1 = A^-1AK
A^-1B^-1 = K
so, (BA)^-1 = A^-1B^-1  {∵ (BA)^-1 = K}

Given: 
The inverse of matrix (A) = adjoint of A / det(A) 

The inverse of 

Hence, The correct option is 3.

Since A is a scalar matrix, we have 

Given matrix 
Now its transpose will be 

The product will be 

⇒ AA^T = I;
∴ (AA^T)^-1 = (I)^-1 = I;

The inverse of a matrix:
The inverse of a matrix A is defined by the following formula.

Where, Adj(A) = [cofactor(A)]^{T
cofactor(A) =}

For a 2 × 2 matrix there is a short-cut formula for obtaining the inverse:


Note: Inverse exists only for a non-singular matrix (i.e. 

Let M⁴ = I      ---(i)
Multiplying by M^-1 on both sides
M⁴ M^-1 = I – M^-1
M³ = M^-1      ---(ii)
Multiplying M⁴ on both sides in equation (i)
M⁴⋅ M⁴ = I ⋅ M⁴
M⁸ = M⁴      ---(iii)
Multiplying M^-1 on both sides, we get:
M⁸ – M^-1 = M⁴ M^-1
M⁷ = M³      ---(iv)
From (ii) and (iv)
M^-1 = M³ = M⁷
Now again multiplying M⁴ on both sides in equation (iii)
M⁸⋅ M⁴ = M⁴ M⁴
M¹² = M⁸
M¹² = M⁴ [from (iii)]
M¹² = I      ---(v) [from (i)]
Multiplying M^-1 on both sides we get
M¹¹ = M^-1      ---(vi)
From (ii), (iv) and (vi)
M^-1 = M³ = M⁷ = M¹¹ = …. & So on
3, 7, 11 ….. from general term 4k + 3 where k = 0, 1, 2
Hence, M^-1 = M^{4k + 3}
Option C is correct.

Determinant of 
Where, Determinant of A = 1(4 × 1 - 5 × 0) - 2(0 - 0) + 3(0 - 0) = 4
Given:
Determinant of A = 4 
So, Determinant of (A^-1) = 

adj.A = Transverse of matrix 
A₁₁ = A_ij = (-1)ⁿ × (cross multiplication of remaining column and row) = (-1)ⁿ × 
Where, n = (i + j)
We follow the same procedure for the A₁₁, A₁₂, A₁₃, .......as so on. Also find the determinant of matrix A. Then applies the given formula after that we finally obtain the A^-1.

Given:

The inverse of matrix (A) = adjoint of A / det(A) 

Calculations:
The inverse of / (Determinant of given matrix)

Hence, The correct option is 3.

Both the given statements are true.