Determinants

The value of $\begin{vmatrix} \sin^2 \theta & \cos^2 \theta \\ -1 & 1 \end{vmatrix}$ is:

Evaluate: $\begin{vmatrix} 1 & \omega \\ \omega & \omega^2 \end{vmatrix}$ where $\omega$ is a complex cube root of unity.

If $A$ is a $3 \times 3$ matrix and $|A| = k$, then $|-2A|$ is:

If $\begin{vmatrix} 3 & 1 \\ x & 2 \end{vmatrix} = \begin{vmatrix} 4 & 1 \\ 2 & 1 \end{vmatrix}$, then $x$ is:

If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, find $|A \cdot A^T|$.

The roots of the equation $\begin{vmatrix} x & 0 & 8 \ 4 & 1 & 3 \ 2 & 0 & x \end{vmatrix} = 0$ are:

If $A$ is a diagonal matrix $diag(d_1, d_2, d_3)$, then $|A|$ is:

Given $|A|=k$ for a $3 \times 3$ matrix $A$, what is $|k A^{-1}|$?

Find the area of the triangle whose vertices are $(a, b+c), (b, c+a),$ and $(c, a+b)$.

If $A$ is a $3 \times 3$ matrix such that $|A| = 4$, then find the value of $|adj(adj A)| / |A|$.

Find the minor $M_{32}$ of the determinant $\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}$.

What is the minor of the element $7$ in the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}$?

Find the area of a triangle with vertices $(3, 8)$, $(-4, 2)$, and $(5, 1)$.

If the vertices of a triangle are $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the area is zero if and only if the points are:

Find the cofactor of the element $a_{21}$ in $\begin{vmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{vmatrix}$.

Given a triangle with vertices $(k, 0)$, $(4, 0)$, and $(0, 2)$. If the area is $4$ square units, find the values of $k$.

If $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$, then $a_{11}A_{31} + a_{12}A_{32} + a_{13}A_{33}$ is equal to:

In a determinant $\Delta = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$, if the cofactor of $b$ is $C_{12} = -(di - fg)$, and the cofactor of $e$ is $C_{22} = (ai - cg)$, what is the result of the expression $a \cdot C_{21} + b \cdot C_{22} + c \cdot C_{23}$?

Let $\Delta$ be the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$. If we define a new triangle with vertices $(x_1+h, y_1+k)$, $(x_2+h, y_2+k)$, and $(x_3+h, y_3+k)$, what is the ratio of the new area to the original area?

If the points $(a, 0)$, $(0, b)$, and $(x, y)$ are collinear, then using the property of determinants for the area of a triangle, which of the following expresses $y$ in terms of $x, a,$ and $b$?

If $A$ is a square matrix of order 4 and $|A| = 2$, then $|\text{adj } A|$ is:

If $A$ is an identity matrix of order $n$, then $|\text{adj } A|$ is:

If $\text{adj } A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$, then $\text{adj }(A^T)$ is:

If $A$ is a square matrix such that $|A| \neq 0$, then $A^{-1}$ is equal to:

If $A$ is a square matrix such that its inverse is given by $A^{-1} = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}$, then the value of the determinant of its adjoint, $|\text{adj } A|$, is equal to:

If $A$ is a non-singular square matrix of order 3 satisfying the matrix equation $A^2 = 2A$, then the value of $|\text{adj } A|$ is:

If $A$ is a square matrix of order 3 and $|A| = k$, then $|\text{adj } A|$ is:

If $A$ is a $3 \times 3$ matrix and $|A| = p$, find the value of $|\text{adj}(pA)|$.

Let $A$ and $B$ be two $3 \times 3$ matrices such that $A$ is non-singular and $AB = O$. What is the value of $B$?

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

The matrix product $A \cdot A^{-1}$ results in:

Identify the element $a_{12}$ in the coefficient matrix $A$ for the system $2x + 5y = 10$ and $3x - y = 4$.

A system of equations $AX = B$ is consistent if $|A| = 2$. How many solutions does it have?

If $A = \begin{bmatrix} k & 2 \\ 3 & 6 \end{bmatrix}$ is singular, what is the value of $k$?

If $|A| = 3$ and $A^{-1} = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}$, what is the matrix $A$?

Solve $2x - y = -2$ and $3x + 4y = 3$. What is the value of $x + y$?

Determine $x, y, z$ for: $x + y + z = 6, 2x + y + 2z = 10, x + y = 3$. Find $z$.

Find the value of $k$ such that the system $x + ky + 3z = 0$, $4x + 3y + kz = 0$, $2x + y + 2z = 0$ has a non-trivial solution.

If the system $x+y+z=5, x+2y+3z=9, x+3y+kz=\mu$ has infinitely many solutions, find $k - \mu$.

Solve the system $\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4$, $\frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1$, $\frac{6}{x} + \frac{9}{y} - \frac{20}{z} = 2$. Find $x+y+z$.

If the determinant of $[2x]$ is $10$, what is the value of $x$?

Find the determinant of the matrix $M = [-1]$.

If $A = [x - 5]$, find $|A|$.

Find the determinant of the matrix $[\sin(\pi/2)]$.

If $A = [\int_0^1 (2x - 1) \, dx]$, find the value of $|A|$.

If $A = [\log_{10} 0.1]$, find the value of $|A|$.

If $A = [x]$ and $|A|$ is the degree of the differential equation $\frac{d^2y}{dx^2} + (\frac{dy}{dx})^3 = 0$, find $x$.

If $M = [k]$ where $k = f'(1)$ for $f(x) = \ln(x^2 + x + 1)$, find $\det(M)$.

Let $B = [b]$ where $b = \int_{1}^{2} e^x dx$. Find $\det(B)$.

If $A = [a]$ is a $1 \times 1$ matrix and $a$ is the minimum value of $f(x) = x^2 - 4x + 7$, find $\det(A)$.

Evaluate $\begin{vmatrix} 1 & a \\ 0 & 1 \end{vmatrix}$.

If $|A| = 5$ and $A$ is a $2 \times 2$ matrix, what is the value of $|3A|$?

Find the value of $\begin{vmatrix} \sin 40^\circ & -\cos 40^\circ \\ \sin 50^\circ & \cos 50^\circ \end{vmatrix}$.

Evaluate the determinant: $\begin{vmatrix} \sqrt{3} & -\sqrt{2} \\ \sqrt{2} & \sqrt{3} \end{vmatrix}$

Evaluate the determinant $\begin{vmatrix} \sin 20^\circ & -\cos 20^\circ \\ \sin 70^\circ & \cos 70^\circ \end{vmatrix}$.

If $A = \begin{bmatrix} \alpha & 2 \\ 2 & \alpha \end{bmatrix}$ and $|A|^2 = 25$, find the positive value of $\alpha$.

Find the positive value of $x$ if $\begin{vmatrix} x & 3 \\ 1 & x \end{vmatrix} = \begin{vmatrix} 5 & 3 \\ 1 & 5 \end{vmatrix}$.

Determine the value of $x$ that satisfies $\begin{vmatrix} 1 & x \\ x & 1 \end{vmatrix} + \begin{vmatrix} 1 & 2 \\ 2 & 1 \end{vmatrix} = 0$.

If $\Delta = \begin{vmatrix} x & y \\ u & v \end{vmatrix}$, find the determinant $\begin{vmatrix} 2x+3u & 2y+3v \\ u & v \end{vmatrix}$ in terms of $\Delta$.

Evaluate $\begin{vmatrix} a+b & a-b \\ a-b & a+b \end{vmatrix}$.

Evaluate $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 8 & 12 \\ 5 & 6 & 7 \end{vmatrix}$.

If $|A| = -5$ for a $3 \times 3$ matrix, find $|-A|$.

Calculate $\begin{vmatrix} 1 & x & y \\ 0 & 2 & z \\ 0 & 0 & 3 \end{vmatrix}$.

Evaluate $\begin{vmatrix} 10 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 10 \end{vmatrix}$.

Evaluate $\begin{vmatrix} \log_3 3 & 0 & 0 \\ 0 & \log_2 4 & 0 \\ 0 & 0 & \log_5 125 \end{vmatrix}$.

If $|A| = 10$, find the value of $|A^T|$ for a $3 \times 3$ matrix.

What is the value of $\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix}$?

If $\Delta_1 = \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \\ \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} 1 & 1 & 1 \\ bc & ca & ab \\ a & b & c \\ \end{vmatrix}$, then which of the following is true?

If $A = [a_{ij}]_{3 \times 3}$ and $C_{ij}$ is the cofactor of $a_{ij}$, and $|A|=5$, find $\sum_{i=1}^3 a_{i1}C_{i1}$.

Find the value of $\begin{vmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ac & bc & c^2+1 \\ \end{vmatrix}$.

Area of a triangle with vertices $(a, 0)$, $(0, b)$, and $(0, 0)$ is:

Find the area of the triangle with vertices $(1, 0)$, $(6, 0)$, and $(4, 3)$ using the formula $\Delta = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$.

Which of the following represents the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$?

If the area of a triangle with vertices $(1, 3)$, $(0, 0)$, and $(k, 0)$ is $3$ sq units, then $k$ is:

A triangle $ABC$ is plotted on a coordinate plane with vertices $A(1, 0)$, $B(6, 0)$, and $C(4, k)$. If the area of triangle $ABC$ is $15$ square units, find the possible value(s) of $k$ using the determinant method.

If the area of a triangle with vertices $(3, 4)$, $(-1, 1)$, and $(x, 2)$ is $3.5$ sq units, find $x$.

The points $(k, 2-2k)$, $(-k+1, 2k)$, and $(-4-k, 6-2k)$ are collinear. Find $k$.

Using determinants, find the value of $k$ such that the points $(k, 2-2k)$, $(1-k, 2k)$, and $(-4-k, 6-2k)$ are collinear.

Find the area of a triangle whose vertices are given by the roots of the equation $z^3 - 1 = 0$ in the complex plane, using the determinant method where $z = x + iy$.