If A is a square matrix and I is an identity matrix such that A^2=A, then A(I-2 A)^3+2 A^3 is equal to : Option 1: I+A Option 2: I+2 ~A Option 3: I-A Option 4: A

Answer: A. Solution: If A is a square matrix and I is an identity matrix such that A^2=A, then A(I-2 A)^3+2 A^3 is equal to: We are given that A^2 = A, which means A is an idempotent matrix. First, let's find A^3 using the given property: A^3 = A^2 * A = A * A = A We'll first find (I-2A)^2: (I-2A)^2 = (I-2A)(I-2A) = I^2 - 2A * I - 2I * A + 4A^2 = I - 2A - 2A + 4A^2 = I - 4A + 4A^2 Since A^2 = A, we have: (I-2A)^2 = I - 4A + 4A = I Now we can calculate (I-2A)^3: (I-2A)^3 = (I-2A)^2(I-2A) = I(I-2A) = I - 2A Next, calculate A(I-2A)^3: A(I-2A)^3 = A(I-2A) = A - 2A^2 Since A^2 = A, we have: A(I-2A)^3 = A - 2A = -A Finally, calculate the complete expression A(I-2A)^3+2A^3: A(I-2A)^3+2A^3 = -A + 2A^3 = -A + 2A = A Therefore, A(I-2A)^3+2A^3 = A

Algebra

Matrices & Determinants

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If $A$ is a square matrix and $I$ is an identity matrix such that $A^{2}=A$ , then $A(I-2 A)^{3}+2 A^{3}$ is equal to :

$\mathrm{I}+\mathrm{A}$

$\mathrm{I}+2 \mathrm{~A}$

$\mathrm{I}-\mathrm{A}$

$\mathrm{A}$

✅ Correct Option: 4

If $A$ is a square matrix and $I$ is an identity matrix such that $A^{2}=A$ , then $A(I-2 A)^{3}+2 A^{3}$ is equal to:

We are given that $A^2 = A$ , which means $A$ is an idempotent matrix.

First, let's find $A^3$ using the given property:

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

We'll first find $(I-2A)^2$ :

$(I-2A)^2 = (I-2A)(I-2A) = I^2 - 2A \cdot I - 2I \cdot A + 4A^2$

$= I - 2A - 2A + 4A^2$

$= I - 4A + 4A^2$

Since $A^2 = A$ , we have:

$(I-2A)^2 = I - 4A + 4A = I$

Now we can calculate $(I-2A)^3$ :

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

Next, calculate $A(I-2A)^3$ :

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

Since $A^2 = A$ , we have:

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

Finally, calculate the complete expression $A(I-2A)^3+2A^3$ :

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

Therefore, $A(I-2A)^3+2A^3 = A$

2025: 30 May Shift 1

The system of equations $x + y - z = 1, 3x + y - 2z = 3, x - y + \lambda z = 1$ has infinite number of solutions if $\lambda$ is equal to

2026: 29 May Shift 1

Match List-I with List-II

List-I (Singular matrix)	List-II (A value of $x$ )
(A) $A=\begin{bmatrix}x & 4 \\ 2 & 2x\end{bmatrix}$	(I) 3
(B) $A=\begin{bmatrix}3 & x \\ 3x & 9\end{bmatrix}$	(II) 4
(C) $A=\begin{bmatrix}x & 1 \\ 1 & x\end{bmatrix}$	(III) 2
(D) $A=\begin{bmatrix}x & 1 \\ 16 & x\end{bmatrix}$	(IV) 1

Choose the correct answer from the options given below:

2023: 15 June Shift 2

If the matrix $A = \begin{bmatrix} x & -2 & -5y \\ 2 & 0 & -9 \\ 10 & 3z & 0 \end{bmatrix}$ is skew-symmetric, then the value of $(2x - 3y + 4z)$ is :

2025: 13 May Shift 1

If A and B are matrices of same order, then $(AB^T - BA^T)$ is always

2025: 27 May Shift 1

Let $A$ be a non-singular square matrix of order $n$ , then

Match List-I with List-II

List-I	List-II
(A) $A(\text{adj} A)$	(I) $\frac{1}{\vert A\vert }$
(B) $\vert \text{adj} A\vert$	(II) $\vert A\vert ^n$
(C) $\vert A^{-1}\vert$	(III) $\vert A\vert I$
(D) $\vert A(\text{adj} A)\vert$	(IV) $\vert A\vert ^{n-1}$

Choose the correct answer from the options given below:

2026: 19 May Shift 1

Which of the following are TRUE?

(A) The inverse of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is $\frac{1}{27}\begin{bmatrix} 7 & 3 \\ -2 & 3 \end{bmatrix}$ .

(B) The determinant of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is 29.

(C) The inverse of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is $\frac{1}{29}\begin{bmatrix} 7 & 3 \\ -2 & 3 \end{bmatrix}$ .

(D) The adjoint of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is $\begin{bmatrix} 7 & 3 \\ -2 & 3 \end{bmatrix}$ .

Choose the correct answer from the options given below:

If AAA is a square matrix and III is an identity matrix such that A2=AA^{2}=AA2=A, then A(I−2A)3+2A3A(I-2 A)^{3}+2 A^{3}A(I−2A)3+2A3 is equal to :

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If $A$ is a square matrix and $I$ is an identity matrix such that $A^{2}=A$ , then $A(I-2 A)^{3}+2 A^{3}$ is equal to :