List of top Mathematics Questions on Transpose of a Matrix

Let P be a square matrix such that P² = I – P. For α, β, γ, δ ∈ N, if P^α + P^β = γI – 29P and P^α – P^β = δI – 13P, then α + β + γ – δ is equal to 

Let B =\(\begin{bmatrix} 1 & 3 & α \\ 1 & 2& 3 \\ α & α & 4 \end{bmatrix}\) , α>2 be the adjoint of a matrix A and |A| = 2, then [α - 2α α] B\(\begin{bmatrix} α \\ -2α  \\ α \end{bmatrix}\)is equal to

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : For matrix A =\(\begin{bmatrix} 1 & 2 \\ 8 & 16 \end{bmatrix}\),rank of A = r(A) = 1
Reason R: For matrix A if |A| = 0, then it implies that rank of matrix A given by r(A) is less than full rank.
In the light of the above statements, choose the most appropriate answer from the options given below:

if A=\(\frac{1}{5! 6! 7!}\begin{bmatrix} 5! & 6! & 7!\\ 6! & 7! & 8! \\ 7! & 8! & 9! \end{bmatrix}\), then |adj(adj(2A))| is equal t

Let D_k=\(\begin{vmatrix} 1 & 2k & 2k-1\\ n & n^2+n+2 & n^2 \\ n & n^2+n & n^2+n+2 \end{vmatrix} \) If \(∑^n_{k=1} D_k=96,\) The n is equal to

Let A=\(\begin{bmatrix} 1 & \frac{1}{51} \\ 0 & 1\end{bmatrix}\). If B=\(\begin{bmatrix} 1 & 2 \\ -1 & -1\end{bmatrix}\)A \(\begin{bmatrix} -1 & -2 \\ 1 & 1\end{bmatrix}\), then the sum of all the elements of the matrix \(\sum^{50}_{n=1}B^n\) is equal to

let \(A=\begin{bmatrix} 1 & \frac{1}{51} \\ 0& 1 \end{bmatrix}\) if \(B=\begin{bmatrix} 1 &2\\ -1& -1 \end{bmatrix} A=\begin{bmatrix} -1 &2\\ 1& 1 \end{bmatrix}\) then the sum of all the elements of the matrix \(∑ ^{50}_{n=1}\) Bⁿ is Equal to 

Find the scalars λ₁ and λ₂ that are attached to \(v=\begin{bmatrix}  1\\ 2 \\  \end{bmatrix} \)and \(w=\begin{bmatrix}  3\\ 1 \\  \end{bmatrix} \) to yield\(\begin{bmatrix}  1\\ 0 \\  \end{bmatrix} \)

Let A = \(\begin{pmatrix} p & -q \\ q & p \end{pmatrix}\) be a 2 × 2 matrix.
A. The rank of matrix A given by r(A) is 3
B. Matrix A is orthogonal if and only if p² + q²=1
C. Transpose of matrix A is given by A^T \(=\begin{pmatrix} p & q \\ -q & p \end{pmatrix}\)
D. Determinant of matrix A is | A |= p
Choose the most appropriate answer from the options given below:

Let A be a \(3×3\) real matrix such that \(A\begin{pmatrix}   1 \\1  \\ 0  \end{pmatrix}\)=\(\begin{pmatrix}   1 \\1  \\ 0  \end{pmatrix}\); \(A\begin{pmatrix}   1 \\0  \\ 1  \end{pmatrix}\)=\(A\begin{pmatrix}   -1 \\0  \\ 1  \end{pmatrix}\) and \(A\begin{pmatrix}   0 \\0  \\ 1  \end{pmatrix}\)=\(\begin{pmatrix}   1 \\1  \\ 2  \end{pmatrix}\)
If \(X = [x_1, x_2, x_3]^T \)and \(I\) is an identity matrix of order \(3\), then the system \([A−2I]X \)= \(\begin{pmatrix}   4 \\1  \\ 1 \end{pmatrix}\) has:

Let A = [a_ij] be a square matrix of order 3 such that a_ij = 2^j–i, for all i, j = 1, 2, 3. Then, the matrix A² + A³ + … + A¹⁰ is equal to 

If $\begin{vmatrix}a+b+2c&a&b\\ c&2a+b+c&b\\ c&a&a+2b+c\end{vmatrix} = 2 $, then $a^3 + b^3 + c^3 - 3abc$ =

If $k$ is one of the roots of the equation $x^2 - 25x + 24 = 0 $ such that $A = \begin{bmatrix}1&2&1\\ 3&2&3\\ 1&1&k\end{bmatrix} $ is a non-singular matrix, then $A^{-1}$ =

Let x be an $n \times 1$ matrix. Let O and I be the zero, and identity matrices of order n, respectively. Define $P = - \frac{xx^T}{x^Tx}$ is the transpose of x. Then which of the following options is always CORRECT?

If $A = \frac{1}{3} \begin{bmatrix}1&2&2\\ 2&1&-2\\ a&2&b\end{bmatrix} $ is an orthogonal matrix, then

The system $2\,x+3 y+z=5,3\,x+y+5\,z=7$ and $x+4\,y-2\,z=3$ has

If $\begin{bmatrix}\alpha&\beta\\ \gamma&-\alpha\end{bmatrix}$ is square root of identity matrix of order $2$ then

If \(\begin{vmatrix}1&-1&x\\1&x&1\\x&-1&1\end{vmatrix}\) has no inverse, then the real value of \(x\) is

If one of the roots of $\begin{vmatrix} 3 &5 & x \\ 7 & x & 7 \\ x & 5 & 3 \end{vmatrix} = 0 $ is -10,then the other roots are

If x = -9 is a root of A = $\begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \\ \end{vmatrix}$ = 0, then other two root are

The identity element in the group $M = \left\{ \begin{bmatrix} x & x \\ x & x\\ \end{bmatrix} | x \ \in \ R, x \neq 0 \right\}$ with respect to matrix multiplication is

Let $A=\left(\begin{array}{ccc}0& 2q& r\\ p& q& -r\\ p& -q& r\end{array}\right)$. If $A{A}^T=I_3$, then $|p|$ is:

If $\begin{bmatrix}1&x&1\end{bmatrix} \begin{bmatrix}1&3&2\\ 2&5&1\\ 15&3&2\end{bmatrix}\begin{bmatrix}1\\ 2\\ x\end{bmatrix} = 0 $, then x can be

If $A=\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}, P=\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}$ and $X=A P A^{T}$, then $A^{T} X^{50} A=$

If f(x)= $\begin{vmatrix} x-3 & 2x^2-18 & 2x^3-81\\ x-5 & 2x^2-50 & 4x^3-500\\ 1 & 2 & 3 \\ \end{vmatrix}$ then f(1). f(3) + f(3). f(5) + f(5). f(1) is