Matrices

If A = $\begin{pmatrix} 1 & 2 \ -2 & -1 \end{pmatrix}$, then which ONE of the following is A⁸ ?

Let L, M, and N be non-singular matrices of order 3 satisfying the equations L^2 = L^-1, M = L^8 and N = L^2. Which ONE of the following is the value of the determinant of (M - N)?

Consider a system of linear equations $PX = Q$ where $P \in \mathbb{R}^{3 \times 3}$ and $Q \in \mathbb{R}^{3 \times 1}$. Suppose $P$ has an LU decomposition, $P = LU$, where $L =...

Consider a system of linear equations $P X=Q$ where $P \in \mathbb{R}^{3 \times 3}$ and $Q \in \mathbb{R}^{3 \times 3}$. Suppose $P$ has an $L U$ decomposition, $P=L U$, where $$L=...

Let $L, M$, and $N$ be non-singular matrices of order 3 satisfying the equations $L^2=L^{-1}, M=L^8$ and $N=L^2$. Which ONE of the following is the value of the determinant of $(M-...

Let A be an n×n matrix over the set of all real numbers R. Let B be a matrix obtained from A by swapping two rows. Which of the following statements is/are TRUE?

Consider solving the following system of simultaneous equations using LU decomposition. x 1 + x 2 $$-$$ 2x 3 = 4 x 1 + 3x 2 $$-$$ x 3 = 7 2x 1 + x 2 $$-$$ 5x 3 = 7 where L and U ar...

Let A and B be two n$$ \times $$n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements,...

Let $P = \begin{bmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{bmatrix}$ and $Q = \begin{bmatrix} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{bmatrix}$ be two matrices. The...

In the LU decomposition of the matrix $$\left[ {\matrix{ 2 & 2 \cr 4 & 9 \cr } } \right]$$, if the diagonal elements of U are both 1, then the lower diagonal entry $${l_{22}}$$ of...

$$\left[ A \right]$$ is a square matrix which is neither symmetric nor skew-symmetric and $${\left[ A \right]^T}$$ is its transpose. The sum and differences of these matrices and d...

How many of the following matrices have an eigen value $$1$$? $$\left[ {\matrix{ 1 & 0 \cr 0 & 0 \cr } } \right],\,\,\left[ {\matrix{ 0 & 1 \cr 0 & 0 \cr } } \right],\,\,\left[ {\m...

Consider the following statements: S1: The sum of two singular n x n matrices may be non-singular S2: The sum of two n x n non-singular matrices may be singular Which of the follow...

Let $$A = \left[ {\matrix{ {{a_{11}}} & {{a_{12}}} \cr {{a_{21}}} & {{a_{22}}} \cr } } \right]\,\,$$ and $$B = \left[ {\matrix{ {{b_{11}}} & {{b_{12}}} \cr {{b_{21}}} & {{b_{22}}}...

The matrices$$\left[ {\matrix{ {\cos \,\theta } & { - \sin \,\theta } \cr {\sin \,\,\theta } & {\cos \,\,\theta } \cr } } \right]\,\,and$$ $$\left[ {\matrix{ a & 0 \cr 0 & b \cr }...

Let A be the set of all nonsingular matrices over real numbers and let * be the matrix multiplication operator. Then

If A and B are real symmetric matrices of size n x n. Then, which one of the following is true?