Algebra

Matrices & Determinants

Medium

common

If A is a square matrix such that $A^2 = A$ and I is the identity matrix of same order as A, then the matrix $(2I+A)^3 - 19A - 3I$ is equal to

✅ Correct Option: 1

Given $A^2 = A$ and $I$ is the identity matrix of the same order as $A$ .

Since $A^2 = A$ :

$A^3 = A^2 \cdot A = A \cdot A = A^2 = A$

So every power of $A$ equals $A$ itself.

Expanding $(2I + A)^3$ using the binomial formula:

$(2I + A)^3 = (2I)^3 + 3(2I)^2(A) + 3(2I)(A^2) + A^3$

$(2I)^3 = 8I$

$3(2I)^2(A) = 3 \cdot 4I \cdot A = 12A$

$3(2I)(A^2) = 6I \cdot A^2 = 6A$ (since $A^2 = A$ )

$A^3 = A$ (since $A^3 = A$ )

So:

$(2I + A)^3 = 8I + 12A + 6A + A$

$= 8I + 19A$

$(2I+A)^3 - 19A - 3I$

$= (8I + 19A) - 19A - 3I$

$= 8I + 19A - 19A - 3I$

$= 5I$

2025: 14 May Shift 1

Let $A = \begin{bmatrix} 1 & 2 & 1 \\ -1 & 3 & 2 \\ 2&4&1\end{bmatrix}$ and $M_{ij}$ , $A_{ij}$ respectively denote the minor, co-factor of an element $a_{ij}$ of matrix A, then which of the following are true?

(A) $M_{22} = -1$

(B) $A_{23} = 0$

(D) $M_{23} = 1$

(E) $M_{32} = -3$

Choose the correct answer from the options given below:

2026: 29 May Shift 2

With reference to the matrix $A=\begin{bmatrix}5&0&0\\0&5&0\\0&0&-5\end{bmatrix}$ , which of the following are TRUE ?

A. |A|=125

B. A is a scalar matrix.

C. A is a diagonal matrix.

D. |A|=-125

Choose the correct answer from the options given below:

2026: 18 May Shift 2

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. All positive integral powers of a symmetric matrix are	I. Unit matrix.
B. Positive odd integral powers of a skew-symmetric matrix are	II. Null matrix.
C. All the matrices which are both symmetric and skew-symmetric, are	III. Skew - Symmetric matrix.
D. All the square matrices whose each diagonal element is one and each non-diagonal element is equal to zero, are	IV. Symmetric matrix.

Choose the correct answer from the options given below:

2025: 3 June Shift 2

If A is an invertible matrix, then which of the following statement(s) is/are TRUE?

(A) $|A^{-1}| = |A|$

(B) $(A^{-1})^{-1} = A$

(D) $(A^T)^{-1} = (A^{-1})^T$

Choose the correct answer from the options given below:

2026: 29 May Shift 1

If $A=\begin{bmatrix} x & p & x+p \\ y & q & y+q \\ z & r & z+r \end{bmatrix}$ , then $|A|$ is

2025: 13 May Shift 2

Let $A = \begin{bmatrix} 2 & -3 & 4 \\ 0 & 1 & 5 \\ -4 & 2 & 3 \end{bmatrix}$ and $a_{ij}$ be any element of matrix A, i, j ∈ {1,2,3}, then which of the following are TRUE?

(A) Minor of $a_{23} = 16$

(B) Minor of $a_{23} = -8$

(D) Cofactor of $a_{23} = 8$

(E) Cofactor of $a_{13} = 4$

Choose the correct answer from the options given below:

If A is a square matrix such that A2=AA^2 = AA2=A and I is the identity matrix of same order as A, then the matrix (2I+A)3−19A−3I(2I+A)^3 - 19A - 3I(2I+A)3−19A−3I is equal to

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If A is a square matrix such that $A^2 = A$ and I is the identity matrix of same order as A, then the matrix $(2I+A)^3 - 19A - 3I$ is equal to