If A is a singular matrix, then Aadj A is equal to Option 1: a unit matrix Option 2: a null matrix Option 3: a symmetric matrix Option 4: a skew symmetric matrix

Answer: a null matrix. Solution: For a singular matrix A, the determinant is zero: (A) = 0. There is a fundamental property relating a matrix to its adjoint: A(adj A) = (A) x I where I is the identity matrix. Since A is singular, (A) = 0. Substituting into the formula: A(adj A) = (A) x I A(adj A) = 0 x I A(adj A) = O where O is the null matrix (a matrix with all elements equal to zero). Therefore, A(adj A) is a null matrix.

Algebra

Matrices & Determinants

Medium

core

If A is a singular matrix, then A{adj A} is equal to

a unit matrix

a null matrix

a symmetric matrix

a skew symmetric matrix

✅ Correct Option: 2

For a singular matrix $A$ , the determinant is zero: $\det(A) = 0$ .

There is a fundamental property relating a matrix to its adjoint:

$A(\text{adj } A) = \det(A) \times I$

where $I$ is the identity matrix.

Since $A$ is singular, $\det(A) = 0$ .

Substituting into the formula:

$A(\text{adj } A) = \det(A) \times I$

$A(\text{adj } A) = 0 \times I$

$A(\text{adj } A) = O$

where $O$ is the null matrix (a matrix with all elements equal to zero).

Therefore, $A(\text{adj } A)$ is a null matrix.

2026: 14 May Shift 2

If $A=\begin{bmatrix}a & 0\\0 & 0\end{bmatrix}$ and $B=\begin{bmatrix}0 & 0\\0 & b\end{bmatrix}$ then the matrix $A^3B^3$ is:

2026: 18 May Shift 2

For the matrix $A=\begin{bmatrix} 0 & 0 & 5 \\ 0 & 6 & 0 \\ 7 & 0 & 0 \end{bmatrix}$ , which of the following statement(s) is/are TRUE ?

A. $A$ is a scalar matrix.

B. $A$ is a diagonal matrix.

C. $A$ is a square matrix.

D. $A$ is a skew symmetric matrix.

Choose the correct answer from the options given below:

2026: 26 May Shift 2

The value/s of ' $\lambda$ ' for which the system of equations

$\lambda x + 3y - z = 1$

$x + 2y + z = 2$

$-\lambda x + y + 2z = -1$

has unique solution is/are:

(where $R$ is set of real numbers)

2022: 25 May Shift 1

If the matrix $A = \begin{bmatrix} 0 & x+y & 1 \\ 3 & z & 2 \\ x-y & -2 & 0 \end{bmatrix}$ is skew-symmetric, then :

2026: 26 May Shift 2

If $A = \begin{bmatrix} 9 & 7 & 5 \\ 6 & 4 & 2 \\ 1 & 3 & 8 \end{bmatrix}$ , then (cofactor of $a_{13}$ - cofactor of $a_{32}$ )will be equal to :

2026: 29 May Shift 1

Let $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ and $I$ be an identity matrix of order 3. If $A^TA=I$ , then $x^2+y^2+z^2$ is equal to:

If A is a singular matrix, then A{adj A} is equal to

Solution

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