Matrices

Two matrices $A_{m \times n}$ and $B_{p \times q}$ can be equal only if:

The zero matrix of order $2 \times 3$ is written as:

For a matrix $A = [a_{ij}]_{4 \times 3}$, what is the maximum value of $i$?

If $A = [a_{ij}]_{3 \times 3}$ is an identity matrix, then $a_{21}$ is:

If $A = [a_{ij}]_{3 \times 2}$ such that $a_{ij} = i + 2j$, then the element $a_{32}$ is:

In a square matrix of order $4$, how many elements lie on the principal diagonal?

For a zero matrix of order $2 \times 3$, how many elements are there?

The total number of elements in a matrix $A = [a_{ij}]_{m \times n}$ where $i \in \{1, 2, 3\}$ and $j \in \{1, 2, 4\}$ is:

Which of the following represents a null matrix?

If $A = [a_{ij}]_{n \times n}$ where $a_{ij} = i+j$, then the trace of $A$ is:

If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, then its transpose $A'$ is:

For any square matrix $A$, the matrix $\frac{1}{2}(A - A')$ is always:

If $A = \begin{bmatrix} 5 & 6 \\ 6 & 5 \end{bmatrix}$, then $A - A'$ is equal to:

If $A = \begin{bmatrix} 0 & a \\ b & 0 \end{bmatrix}$ is a skew-symmetric matrix, then the relationship between $a$ and $b$ is:

If $A$ is a skew-symmetric matrix of order $n \times n$, then the elements $a_{ij}$ satisfy:

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

If $A$ is a square matrix, $A^T$ is its transpose, and $A = -A^T$, then $A$ is:

Let $A$ be a $3 \times 3$ symmetric matrix with all entries as 1. What is the rank of $A - A^T$?

If $A$ and $B$ are symmetric matrices of the same order, then $ABA + BAB$ is:

Let $A$ be a $3 \times 3$ matrix such that $A^T = -A$. What is $\det(A^{2023})$?

If X + $$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, then $X$ is:

If $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$, then $AB$ is:

If $A = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 4 \end{bmatrix}$, the order of $AB$ is:

If $A = \begin{bmatrix} 2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 4 \end{bmatrix}$, the order of $A + B$ is:

If $A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$, then $A^2 - 3I$ is equal to:

If $A$ is a $3 \times 2$ matrix, $B$ is a $2 \times 4$ matrix, and $C$ is a $4 \times 1$ matrix, the order of the matrix $ABC$ is:

If $\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}$\begin{bmatrix} x \ y \end{bmatrix}$= \begin{bmatrix} 1 \\ 4 \end{bmatrix}$, then the values of $x$ and $y$ are:

If $A = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$, where $i^2 = -1$, find $A^2$.

Find the matrix $X$ such that $2A + B + X = O$, where $A = \begin{bmatrix} -1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & -2 \\ 1 & 5 \end{bmatrix}$.

If $A$ is a $3 \times 3$ matrix such that $A^2 = A$, find the trace of $A$ if it is known that $A$ is not the identity matrix and not the null matrix, and all its eigenvalues are integers.

If $A = [a_{ij}]$ is a square matrix such that $a_{ij} = 0$ for $i \neq j$, it is called a:

A matrix having only one column is called a:

If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, then $A + A$ is:

Matrix multiplication $AB$ is defined if:

The law expressed by the equation $(A + B)C = AC + BC$ is the:

If $A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$, then $A^2$ is equal to:

The total number of possible $2 \times 3$ matrices with each entry being $1$ or $-1$ is:

If $A$ and $B$ are square matrices of the same order, which of the following is always true for the transpose operation?

If $A = \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}$, then $A^n$ is equal to:

If $A = \text{diag}(d_1, d_2, d_3)$, then $A^n$ is:

The uniqueness of the inverse theorem states that if a matrix has an inverse, it has:

If a square matrix $A$ has an inverse $B$, then the product of their determinants $|A| \cdot |B|$ is:

If $A$ is an invertible matrix and $k = -1$, then $(-A)^{-1}$ is equal to:

Which of the following symbols is commonly used to denote the inverse of a matrix $A$?

If $A$ is an invertible matrix of order 2 such that its determinant $|A| = 5$, then the determinant of its inverse $|A^{-1}|$ is:

For any invertible matrix $A$ and a natural number $n$, $(A^n)^{-1}$ is equal to:

When finding the inverse of a matrix $A$ using elementary row operations, we start with the equation $A = IA$. If we transform the left side $A$ into $I$ using a sequence of operations, the same operations transform the $I$ on the right side into:

If $A = \begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix}$, find $(A^2)^{-1}$.

If $A$ is an invertible matrix and $B$ is a matrix such that $AB = O$, then $B$ must be:

Let $A$ be a non-singular matrix of order 3 such that $A^2 - 4A + I = O$. The value of $det(A - 4I + A^{-1})$ is:

For a matrix $[a_{ij}]_{2 \times 3}$, what is the maximum value $j$ can take?

If a matrix $A$ has 8 elements, which of these is NOT a possible order?

What is the order of a matrix with $m$ rows and $n$ columns?

If a matrix has 1 element, its order must be:

If $A$ is a matrix of order $2 \times 3$ and $B$ is a matrix of order $3 \times 2$, what is the order of the matrix $2A - 3B^T$?

How many possible orders can a matrix have if it contains 21 elements?

A matrix $A = [a_{ij}]_{m \times n}$ has all elements equal to 3. If the sum of all elements is 36 and $m = 4$, what is the order of the matrix?

If the number of elements in a matrix is $20$, what is the number of all possible orders $m \times n$ such that $m$ and $n$ are even?

If $A = [a_{ij}]$ is a $3 \times 3$ matrix such that $a_{ij} = 1$ if $i \le j$ and $a_{ij} = 0$ if $i > j$, what is the sum of all elements of $A$?

How many elements are in a diagonal matrix of order $n$ that are necessarily zero?

Two matrices $A$ and $B$ are equal. If the order of $A$ is $2 \times 3$, what must be the order of $B$?

If $\begin{bmatrix} x^3 \end{bmatrix} = \begin{bmatrix} 8 \end{bmatrix}$, find the real value of $x$.

If $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} x - 1 \\ y - 2 \\ z - 3 \end{bmatrix}$, find the value of $x$.

If $\begin{bmatrix} 5x & 0 \end{bmatrix} = \begin{bmatrix} 20 & 0 \end{bmatrix}$, find the value of $x$.

Find $x$ and $y$ if $\begin{bmatrix} x+1 & 2 \\ 3 & y-1 \end{bmatrix} = \begin{bmatrix} 2x-3 & 2 \\ 3 & 3y-7 \end{bmatrix}$.

Determine $x$ and $y$ if $\begin{bmatrix} 2x & x+y \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 4 & 5 \\ 3 & 4 \end{bmatrix}$.

If $\begin{bmatrix} x+3y & 4 \\ 1 & x-y \end{bmatrix} = \begin{bmatrix} 7 & 4 \\ 1 & 3 \end{bmatrix}$, find $x$.

If \begin{bmatrix} x+y & z+w \\ x-y & z-w \\end{bmatrix} = \begin{bmatrix} 5 & 7 \\ 1 & 3 \\end{bmatrix}, find the value of $x+y+z+w$.

If \begin{bmatrix} x+y & 3 \\ 4 & xy \\end{bmatrix} = \begin{bmatrix} 6 & 3 \\ 4 & 8 \\end{bmatrix}, find $|x-y|$.

If \begin{bmatrix} x+10 & y^2+2y \\ 0 & -4 \\end{bmatrix} = \begin{bmatrix} 3x+4 & 3 \\ 0 & -4 \\end{bmatrix}, find the set of possible values for $y$.

Is the sum of two $2 \times 2$ identity matrices equal to the $2 \times 2$ identity matrix?

If $A + B = C$, then $C - B$ is equal to:

What is the element in the 2nd row and 1st column of the sum of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$?

If the order of matrix $A$ is $4 \times 5$ and the order of matrix $B$ is $4 \times 5$, then the order of matrix $3A + 2B$ is:

If $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, what is $A + A^T$?

Find the value of $x + y$ if $\begin{bmatrix} x \\ y \end{bmatrix} +$\begin{bmatrix} y \ x \end{bmatrix}$= \begin{bmatrix} 4 \\ 4 \end{bmatrix}$.

If $A$ is any matrix, then $A + O = O + A = A$. This property is known as:

Which of the following is the additive inverse of the matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

If $A = [1 \quad 0]$ and $B = [0 \quad 1]$, find $2A + 3B$.

Let $A$ be a $3 \times 3$ matrix. Find the value of $(A + A^T) - (A^T + A)$.

If $k(A+B) = kA + kB$, and $k=0$, the result is always:

If $A$ is a null matrix, then $100A$ is:

Given $A = \begin{bmatrix} 2 & -2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 1 \end{bmatrix}$, find $2A + 4B$.

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