Matrices and Determinants

Which of the following is correct?

If A and B are invertible square matrices of order n, then which of the following is not correct?

The system of equations $x + 2y = 3$ and $2x + 3y = 3$ has

Match List-I with List-IIList-IList-IIa)A matrix which is not a square matrixi)Symmetric matrixb)A square matrix $A = A'$ii)Null matrixc)The diagonal elements of a diagonal matrix are sameiii)Rectangular matrixd)A matrix which is both symme...

Consider the following statements:Statement I: If A is a non-singular matrix, then $A^{-1}$ exists.Statement II: If A and B are symmetric matrices of same order, then $(AB - BA)$ is a skew symmetric matrix.Choose the correct option.

A row matrix has only

Let X be a matrix of order $2 \times n$ and Z be a matrix of order $2 \times p$. If $n = p$, then the order of the matrix $8X - 9Z$ is:

If A and B are invertible matrices of same order, then which of the following is not correct?

The area of the triangle with vertices $(3, 8), (-4, 2)$ and $(5, 1)$ is $\dfrac{P}{4}$, then the value of $P$ is

If $B=\left[\begin{array}{ll}1 & 3 \\ 1 & \alpha\end{array}\right]$ be the adjoint of a matrix $A$ and $|A|=2$, then the value of $\alpha$ is

If $A$ is a square matrix of order $3 \times 3, \operatorname{det} A=3$, then the value of $\operatorname{det}\left(3 A^{-1}\right)$ is

If $A$ is a square matrix satisfying the equation $A^2-5 A+7 I=0$, where $I$ is the $I$ dentity matrix and 0 is null matrix of same order, then $A^{-1}=$

If $A=\left[\begin{array}{ll}k & 2 \\ 2 & k\end{array}\right]$ and $\left|A^3\right|=125$, then the value of $k$ is

If a matrix $A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$ satisfies $A^6=k A^{\prime}$, then the value of $k$ is

Which of the following statements is not correct?

If $A$ and $B$ are two matrices such that $A B$ is an identity matrix and the order of matrix $B$ is $3 \times 4$, then the order of matrix $A$ is

If $A$ is a square matrix such that $A^2=A$, then $(I-A)^3$ is

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

If $f(x)=\left|\begin{array}{ccc}x-3 & 2 x^2-18 & 2 x^3-81 \\ x-5 & 2 x^2-50 & 4 x^3-500 \\ 1 & 2 & 3\end{array}\right|$, then $f(\mathrm{l}) \cdot f(3)+f(3) \cdot f(5)+f(5) \cdot f(\mathrm{l})$ is

If $A$ is a square matrix, such that $A^2=A$, then $(I+A)^3$ is equal to

If $A=\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right)$, then $A^{10}$ is equal to

If $A=\left|\begin{array}{cc}x & 1 \\ 1 & x\end{array}\right|$ and $B=\left|\begin{array}{ccc}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{array}\right|$, then $\frac{d B}{d x}$ is

If $$A=\left[\begin{array}{cc}1 & \tan \alpha / 2 \\ -\tan \alpha / 2 & 1\end{array}\right]$$ and \(A B=I\), then \(B\) is equal to

If $$\Delta=\left|\begin{array}{ccc}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{array}\right|$$ and $$\Delta_1=\left|\begin{array}{ccc}1 & 1 & 1 \\ b c & c a & a b \\ a & b & c\end{array}\right|$$, then

If $$A=\left[\begin{array}{cc}2-k & 2 \\ 1 & 3-k\end{array}\right]$$ is singular matrix, then the value of \(5 k-k^2\) is equal to

If \(A\) and \(B\) are two matrices, such that \(A B=B\) and \(B A=A\), then \(A^2+B^2\) equals to

If $$x\left[\begin{array}{l}3 \\ 2\end{array}\right]+y\left[\begin{array}{r}1 \\ -1\end{array}\right]=\left[\begin{array}{l}15 \\ 5\end{array}\right]$$, then the value of \(x\) and \(y\) are

$$\text { The value of }\left|\begin{array}{ccc}
\sin ^2 14^{\circ} & \sin ^2 66^{\circ} & \tan 135^{\circ} \\
\sin ^2 66^{\circ} & \tan 135^{\circ} & \sin ^2 14^{\circ} \\
\tan 135^{\circ} & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}
\end{arr...

If $$A_n=\left[\begin{array}{cc}1-n & n \\ n & 1-n\end{array}\right]$$, then
\(\left|A_1\right|+\left|A_2\right|+\ldots .\left|A_{2021}\right|=\)

If there are two values of '\(a\)' which makes determinant
$$\Delta=\left|\begin{array}{ccc} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a \end{array}\right|=86$$
Then, the sum of these number is

If \(A\) is a \(3 \times 3\) matrix such that \(|5 \cdot \operatorname{adj} A|=5\), then \(|A|\) is equal to

If $$A=\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]$$, then \((a I+b A)^n\) is (where \(I\) is the identify matrix of order 2)

If \(A\) is a skew symmetric matrix, then A\(^{2021}\) is

If $$A=\left[\begin{array}{ll}2 & -1 \\ 3 & -2\end{array}\right]$$, then the inverse of the matrix \(A^3\) is

If \(A\) is a matrix of order \(3 \times 3\), then \(\left(A^2\right)^{-1}\) is equal to

If \(x^3-2 x^2-9 x+18=0\) and $$A=\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & x & 6 \\ 7 & 8 & 9\end{array}\right|$$ then the maximum value of \(A\) is

If \(A\) and \(B\) are matrices of order 3 and \(|A|=5,|B|=3\), then \(|3 A B|\) is

If \(A\) and \(B\) are invertible matrices then which of the following is not correct?

Let \(M\) be \(2 \times 2\) symmetric matrix with integer entries, then \(M\) is invertible if

If $$A=\left[\begin{array}{ccc}1 & -2 & 1 \\ 2 & 1 & 3\end{array}\right]$$
$$ B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 1\end{array}\right]$$, then \((A B)^{\prime}\) is equal to

If \(A\) is a square matrix of order 3 and \(|A|=5\), then \(\mid A\) adj. \(A \mid\) is

If $$f(x)=\left|\begin{array}{ccc}x^3-x & a+x & b+x \\ x-a & x^2-x & c+x \\ x-b & x-c & 0\end{array}\right|$$, then

If $$A=\left(\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right)$$ then \(A^4\) is equal to

If \(A\) and \(B\) are square matrices of same order and \(B\) is a skew symmetric matrix, then \(A^{\prime} B A\) is

If \(a_1 a_2 a_3 \ldots a_9\) are in AP, then the value of $$\left|\begin{array}{lll}a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9\end{array}\right|$$ is

If $$\left(\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right) A=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$$ then the matrix \(a\) is

The constant term in the expansion of $$\left|\begin{array}{ccc}3 x+1 & 2 x-1 & x+2 \\ 5 x-1 & 3 x+2 & x+1 \\ 7 x-2 & 3 x+1 & 4 x-1\end{array}\right|$$ is

If the value of a third order determinant is 16, then the value of the determinant formed by replacing each of its elements by its cofactor is

If $$A=\left[\begin{array}{ll}1 & 3 \\ 4 & 2\end{array}\right], B=\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]$$, Then \(\left|A B B^{\prime}\right|=\)

If $$3 A+4 B^{\prime}=\left[\begin{array}{ccc}7 & -10 & 17 \\ 0 & 6 & 31\end{array}\right]$$ and $$2 B+3 A^{\prime}\left[\begin{array}{cc}-1 & 18 \\ 4 & 0 \\ -5 & -7\end{array}\right]$$ then \(B=\)