Algebra

What is the result of $A + A$ if $A$ is a skew-symmetric matrix?

If $A$ is a square matrix such that $A^T = A$, then $A$ is called:

The diagonal elements of a matrix $A = [a_{ij}]$ are those for which:

Every identity matrix is also a:

If $A$ and $B$ are symmetric matrices of the same order, then the product $AB$ is a symmetric matrix if and only if:

If $A$ is a square matrix, then $A + A^T$ is always:

Number of diagonal elements in a square matrix of order $n \times n$ is:

What is the sum of all elements of a skew-symmetric matrix?

For any square matrix $A$, the matrix $A^T A - A A^T$ is always:

The determinant $\det(A - A^T)$ is always 0 if the order of the square matrix $A$ is:

Find the product $\begin{bmatrix} 4 & 5 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix}$.

If $A$ and $B$ are square matrices of the same order, then $k(A + B)$ is equal to:

Find the result of $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.

If $A$ is a $2 \times 2$ matrix such that $A + B = I$, then $B$ is equal to:

If $A = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 0 \ 1 & 1 \end{bmatrix}$, find $AB - BA$.

If $A$ is a square matrix such that $A^2 = I$, then what is the value of $(A-I)^3 + (A+I)^3 - 7A$?

Find the value of $x$ for which \begin{bmatrix} x & 1 \end{bmatrix} $$\begin{bmatrix} 1 & 0 \ -2 & -3 \end{bmatrix}$$ $$\begin{bmatrix} x \ 3 \end{bmatrix}$$ = [0].

If $A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix}$, then $BA$ is a matrix of order:

If $A = \begin{bmatrix} λ & 1 \\ 0 & λ \end{bmatrix}$, find the trace of $A^n$.

If $A$ is a square matrix such that $A^2 = I$, then $A^n = I$ if:

What is the co-factor $C_{22}$ in the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 0 & 4 & 0 \\ 5 & 6 & 7 \end{vmatrix}$?

Find the value of $\begin{vmatrix} \sec^2 \theta & \tan^2 \theta \\ 1 & 1 \end{vmatrix}$.

If $\Delta = \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix}$, then $\Delta$ is equal to which of the following?

Evaluate the determinant: $\begin{vmatrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{vmatrix}$.

If $a, b, c$ are the roots of the cubic equation $x^3 + px^2 + q = 0$, then the value of the determinant $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ is:

If $f(x) = \begin{vmatrix} x & 1 & 1 \ 1 & x & 1 \ 1 & 1 & x \end{vmatrix}$, then $f(1)$ is:

The determinant $\begin{vmatrix} 1 & a & a^2 \ 1 & b & b^2 \ 1 & c & c^2 \end{vmatrix}$ is equal to:

If $p, q, r$ are the roots of the equation $x^3 - ax^2 + bx - c = 0$, find the value of the determinant $\begin{vmatrix} p & q & r \\ q & r & p \\ r & p & q \end{vmatrix}$ in terms of $a$ and $b$.

Let $f(x) = \begin{vmatrix} \cos(x+\alpha) & \cos(x+\beta) & \cos(x+\gamma) \\ \sin(x+\alpha) & \sin(x+\beta) & \sin(x+\gamma) \\ \sin(\beta-\gamma) & \sin(\gamma-\alpha) & \sin(\alpha-\beta) \end{vmatrix}$. Which of the following is true for $f(x)$?

Evaluate the determinant $\Delta = \begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}$ where $A, B, C$ are the angles of a triangle with $A=B=C$.

If $A = \begin{bmatrix} 4 & 1 \\ 3 & 1 \end{bmatrix}$, what is the value of $|A^{-1}|$?

If $A$ is a square matrix of order 2 and $|A| = 5$, then $|adj(adj A)|$ is:

If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then $adj A$ is:

The condition for a matrix $A$ to have an inverse is that $A$ must be:

Given $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, find the trace of the matrix $A \cdot (adj A)$.

If $A$ is a $3 \times 3$ matrix and $|A| = 2$, then $|adj(adj(adj A))|$ is:

Find the adjoint of $A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 4 & 1 \end{bmatrix}$.

If $A$ and $B$ are $2 \times 2$ matrices such that $|A| = 2$ and $|B| = -3$, then $|adj(2AB)|$ is:

If $A$ is a $3 \times 3$ matrix and $A^T A = 4I$, find $|adj A|$.

If $A$ is a non-singular matrix of order 3, and $|adj A| = |A|^k$, then $k$ equals:

In the matrix form of $x + y + z = 1$, $x + y + z = 2$, $x + y + z = 3$, the matrix $A$ would be:

Which of the following matrices has no inverse?

If $A = \begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix}$, then $|A|$ is:

For the system $x + y = 10$, $x - y = 2$, what is the matrix product $AX$?

Find the value of $z$ such that $x + y + z = 9, 2x + 5y + 7z = 52, 2x + y - z = 0$.

Solve for $y$ in the system $x + y + z = 2, 2x - y + z = 5, x + y - z = 0$.

If A $$\begin{bmatrix} x \\ y \end{bmatrix}$$ = \begin{bmatrix} 1 \\ 2 \end{bmatrix} and $A^{-1} = \begin{bmatrix} 3 & -2 \\ -1 & 1 \end{bmatrix}$, find $x$.

Given $A = \begin{pmatrix} 2 & 3 & 10 \\ 4 & -6 & 5 \\ 6 & 9 & -20 \end{pmatrix}$, find $A^{-1}$. Use this to solve $2x + 3y + 10z = 4, 4x - 6y + 5z = 1, 6x + 9y - 20z = 2$. What is $x$?

Find the values of $a$ and $b$ so that the system of equations $x + y + z = 6$, $x + 2y + 3z = 10$, $x + 2y + az = b$ has a unique solution.

For the system $2x - y + 3z = 9, x + y + z = 6, x - y + z = 2$, find the value of $x y z$.