Algebra

Matrices & Determinants

Medium

applied

For $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ , if $X$ and $Y$ are square matrices of order $2$ such that $X Y=X$ and $Y X=Y$ , then $\left(\mathrm{Y}^{2}+2 \mathrm{Y}\right)$ equals to :

$2 Y$

$I+3 X$

$I+3 Y$

$3 \mathrm{Y}$

✅ Correct Option: 4

$I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ (Identity matrix)

$X$ and $Y$ are $2 \times 2$ matrices

$XY = X$ ... (1)

$YX = Y$ ... (2)

From equation (2): $YX = Y$

Multiply both sides by $Y$ on the right:

$YXY = Y^2$

Now, from equation (1): $XY = X$

Multiply both sides by $Y$ on the left:

$Y(XY) = YX$

$YX = YX$ ✓

But we know from equation (2) that $YX = Y$ , so:

$YX = Y$

Let's find $Y^2$ directly.

From $YX = Y$ , multiply both sides on the right by $Y$ :

$YXY = Y^2$ ... (3)

Now, take $YXY$ and use equation (1):

We can rewrite: $YXY = Y(XY)$

Since $XY = X$ from equation (1):

$Y(XY) = YX$

Since $YX = Y$ from equation (2):

$YX = Y$

Therefore, from equation (3):

$Y^2 = YXY = YX = Y$

This means $Y^2 = Y$ (Y is idempotent)

Now we can calculate $Y^2 + 2Y$ :

Since $Y^2 = Y$ :

$Y^2 + 2Y = Y + 2Y = 3Y$

Final Answer: $3Y$

Note: The answer is $3Y$ because $Y$ can be any matrix satisfying the given conditions (like $Y = I$ or $Y = O$ ).

2026: 15 May Shift 1

Match the LIST-I with LIST-II

	LIST-I (Matrix $A =$ )		LIST-II ( $\vert adj A \vert =$ )
A.	$\begin{bmatrix}2 & 1\\ 1 & 3\end{bmatrix}$	I.	$3$
B.	$\begin{bmatrix}1 & 0\\ 3 & 2\end{bmatrix}$	II.	$2$
C.	$\begin{bmatrix}3 & 0\\ 0 & 1\end{bmatrix}$	III.	$5$
D.	$\begin{bmatrix}0 & -1\\ 4 & 0\end{bmatrix}$	IV.	$4$

Choose the correct answer from the options given below:

2025: 2 June Shift 1

If $A$ is a skew-symmetric matrix of order 5, then $|adjA|$ is equal to

2025: 13 May Shift 2

Which of the following statements is (are) true?

(A) $B^T AB$ is a skew-symmetric matrix if A is a symmetric matrix

(B) $B^T AB$ is a symmetric matrix if A is a symmetric matrix

(D) $B^T AB$ is a skew-symmetric matrix if B is a skew-symmetric matrix

(E) $B^T AB$ is a symmetric matrix if B is a symmetric matrix

Choose the correct answer from the options given below:

2026: 21 May Shift 1

If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 3 & -1 \end{bmatrix}$ , then $A^2$ is equal to

2025: 29 May Shift 2

$A = \begin{bmatrix} 1/3 & 2 \\ 0 & 2x - 3 \end{bmatrix}$ & $B = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix}$ If $AB = I$ (Where I is an identity matrix of order 2) , then value of x is

2026: 19 May Shift 2

If A and B are square matrices of same order such that, $AB=BA$ , $A^{-1} = A^2$ and $B^{-1} = B^3$ , then the value of $(AB)^{2040}$ , is

For I=[1001]I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]I=[10​01​], if XXX and YYY are square matrices of order 222 such that XY=XX Y=XXY=X and YX=YY X=YYX=Y, then (Y2+2Y)\left(\mathrm{Y}^{2}+2 \mathrm{Y}\right)(Y2+2Y) equals to :

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

For $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ , if $X$ and $Y$ are square matrices of order $2$ such that $X Y=X$ and $Y X=Y$ , then $\left(\mathrm{Y}^{2}+2 \mathrm{Y}\right)$ equals to :