Recent questions tagged linear-algebra

Matrix $\text{A}$ has the eigenvalues $1,2$, and $3$. The Trace of $\mathrm{A}^{2}$ is$6$$14$$20$$8$

The eigenvalues of $[A]=\left[\begin{array}{ccc}2 & -3.5 & 6 \\ 3.5 & 5 & 2 \\ 8 & 1 & 8.5\end{array}\right]$ are $\lambda_{1}=-1.547, \lambda_{2}=12.330$, and $\lambda_{...

For the matrix $[A]$ given below, the transpose is $\_\_\_\_\_\_\_\_\_\_\_$$$[A]=\left[\begin{array}{lll} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{array}\right]$$$\left[\...

Pick the CORRECT eigen value(s) of the matrix $[A]$ from the following choices.$$[A]=\left[\begin{array}{ll} 6 & 8 \\ 4 & 2 \end{array}\right]$$$10$$4$$-2$$-10$

​​​​​Suppose $\lambda$ is an eigenvalue of matrix $A$ and $x$ is the corresponding eigenvector. Let $x$ also be an eigenvector of the matrix $B=A-2 I$, where $I$ is the i...

​​​​Let $A=\left[\begin{array}{cc}1 & 1 \\ 1 & 3 \\ -2 & -3\end{array}\right]$ and $b=\left[\begin{array}{l}b_{1} \\ b_{2} \\ b_{3}\end{array}\right]$. For $A x=b$ to be ...

Consider two matrices $\mathbf{A}=\left[\begin{array}{lll}2 & 1 & 4 \\ 1 & 0 & 3\end{array}\right]$ and $\mathbf{B}=\left[\begin{array}{cc}-1 & 0 \\ 2 & 3 \\ 1 & 4\end{ar...

For the matrix$[A]=\left[\begin{array}{ccc}1 & -1 & 0 \\-1 & 2 & -1 \\0 & -1 & 1\end{array}\right]$which of the following statements is/are TRUE?$[A]\{x\}=\{b\}$ has a un...

Two vectors $\left[\begin{array}{llll}2 & 1 & 0 & 3\end{array}\right]^{T} $ and $\left[\begin{array}{cccc}1 & 0 & 1 & 2\end{array}\right]^{T}$ belong to the null space of...

Cholesky decomposition is carried out on the following square matrix $[A]$.$$[A]=\left[\begin{array}{cc}8 & -5 \\-5 & a_{22}\end{array}\right]$$Let $l_{\mathrm{ij}}$ and ...

For the matrix$$[A]=\left[\begin{array}{lll}1 & 2 & 3 \\3 & 2 & 1 \\3 & 1 & 2\end{array}\right]$$which of the following statements is/are TRUE?The eigenvalues of $[A]^{T}...

$\text{P}$ and $\text{Q}$ are two square matrices of the same order. Which of the following statement(s) is/are correct?If $\text{P}$ and $\text{Q}$ are invertible, then ...

Let $y$ be a non-zero vector of size $2022 \times 1.$ Which of the following statement(s) is/are $\text{TRUE}?$$yy^{T}$ is a symmetric matrix.$y^{T}y$ is an eigenvalue of...

The matrix $M$ is defined as $$M = \begin{bmatrix} 1 & 3\\ 4 & 2 \end{bmatrix}$$ and has eigenvalues $5$ and $-2$. The matrix $Q$ is formed as $$Q = M^{3} - 4M^{2} - 2M$$...

The rank of the matrix $\begin{bmatrix} 5 & 0 & -5 & 0\\ 0 & 2 & 0 & 1\\ -5 & 0 & 5 & 0\\ 0 & 1 & 0 & 2 \end{bmatrix}$ is$1$$2$$3$$4$

The rank of matrix $\begin{bmatrix} 1 & 2 & 2 & 3\\ 3 & 4 & 2 & 5\\ 5 & 6 & 2 & 7\\ 7 & 8 & 2 & 9 \end{bmatrix}$ is$1$$2$$3$$4$

If $A = \begin{bmatrix} 1 & 5 \\ 6 & 2 \end{bmatrix}$ and $B= \begin{bmatrix} 3 & 7 \\ 8 & 4 \end{bmatrix}, \: AB^T$ is equal to$\begin{bmatrix} 38 & 28 \\ 32 & 56 \end{b...