Matrices — JEE Main PYQs

If $A=\left[\begin{array}{cc}5a& -b\\ 3& 2\end{array}\right]$ and A adj $A=A{A}^T$, then $5a+b$ is equal to:

If $\mathrm{A}=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ \mathrm{a}& 2& \mathrm{b}\end{array}\right]$ is a matrix satisfying the equation ${\mathrm{AA}}^{\mathrm{T}}=9\mathrm{I}$, where I is $3\times 3$ identity matrix, then ordered pair $(\mathrm{a},\mathrm{b})$ is …

The set of all values of $\lambda$ for which the system of linear equations : …

If A is an $3\times 3$ non-singular matrix such that $A{A}^{\prime }=A^{\prime }A$ and $B=A^{-1}{A}^{\prime }$, then $B{B}^{\prime }$ equals

If $P=\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$ is the adjoint of a $3\times 3$ matrix $A$ and $|A|=4$, then $\alpha$ is equal to

The number of values of $k$, for which the system of equations
$(k+1)x+8y=4k$
$kx+(k+3)y=3k-1$
has no solution, is

Let $A=\left(\begin{array}{lll}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right)$. If $u_1$ and $u_2$ are column matrices such that $A{u}_1=\left(\begin{array}{l}1\\ 0\\ 0\end{array}\right)$ and $A{u}_2=\left(\begin{array}{l}0\\ 1\\ 0\end{array}\right)$, t …

Let $P$ and $Q$ be $3\times 3$ matrices with $P\ne Q$. If $P^3=Q^3$ and $P^{2}Q=Q^{2}P$, then determinant of ( $P^2+Q^2$ ) is equal to

Let A and B be two symmetric matrices of order 3.

Statement-1: A(BA) and (AB)A are symmetric matrices.

Statement-2: AB is symmetric matrix if matrix multiplication of A and B is commutative.

The number of $3\times 3$ non-singular matrices, with four entries as 1 and all other entries as 0, is

Let A be a $2\times 2$ matrix with non-zero entries and let $A^2=I$, where I is $2\times 2$ identity matrix. Define
$\mathrm{Tr}(A)=$ sum of diagonal elements of A and $|A|=$ determinant of matrix A.

Statement-1: $\mathrm{Tr}(A)=0$
Statement-2: $|A|=1$

Consider the system of linear equations:
$x_1+2x_2+x_3=3$
$2x_1+3x_2+x_3=3$
$3x_1+5x_2+2x_3=1$
The system has

Let $A$ be a $2\times 2$ matrix

Statement-1: $\text{adj}(\text{adj }A)=A$
Statement-2: $|\text{adj }A|=|A|$

Let A be a $2\times 2$ matrix with real entries. Let I be the $2\times 2$ identity matrix. Denote by $\mathrm{tr}(A)$, the sum of diagonal entries of $A$. Assume that $A^2=I$.
Statement -1: If $A\ne I$ and $A\ne -I$, then $\det A=-1$.
Statement -2 : If …

Let $\mathrm{a},\mathrm{b},\mathrm{c}$ be any real numbers. Suppose that there are real numbers $\mathrm{x},\mathrm{y},\mathrm{z}$ not all zero such that $\mathrm{x}=cy+bz,y=az+cx$ and $z=bx+ay$. Then $a^2+b^2+c^2+2abc$ is equal to

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

If $A$ and $B$ are square matrices of size $n\times n$ such that $A^2-B^2=(A-B)(A+B)$, then which of the following will be always true?

Let $\mathrm{A}=\left(\begin{array}{ll}1& 2\\ 3& 4\end{array}\right)$ and $\mathrm{B}=\left(\begin{array}{ll}\mathrm{a}& 0\\ 0& \mathrm{b}\end{array}\right),\mathrm{a},\mathrm{b}\in \mathrm{N}$. Then

If $A^2-A+I=0$, then the inverse of $A$ is

The system of equations
$\alpha x+y+z=\alpha -1$,
$x+\alpha y+z=\alpha -1$,
$x+y+\alpha z=\alpha -1$
has no solution, if $\alpha$ is

Let $A=\left(\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right)$. The only correct statement about the matrix $A$ is

Let $A=\left(\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right)$ and $B=\left(\begin{array}{ccc}4& 2& 2\\ -5& 0& \alpha \\ 1& -2& 3\end{array}\right)$. If $B$ is the inverse of matrix $A$, then $\alpha$ is