Vardaan Learning Institute

Evaluate the determinant: $\Delta = \begin{vmatrix} 4 & 5 \\ -2 & 3 \end{vmatrix}$.

Find the value of $\begin{vmatrix} \cos 60^\circ & \sin 60^\circ \\ -\sin 60^\circ & \cos 60^\circ \end{vmatrix}$.

If $\begin{vmatrix} 2x & 5 \\ 8 & x \end{vmatrix} = \begin{vmatrix} 6 & 5 \\ 8 & 3 \end{vmatrix}$, find the value(s) of $x$.

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

Evaluate the determinant: $\Delta = \begin{vmatrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}$. (Hint: Expand along a row/column with maximum zeros).

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

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

Solve for $x$: $\begin{vmatrix} x-2 & -3 \\ 3x & 2x \end{vmatrix} = 3$.

What is the determinant of a $3 \times 3$ upper triangular matrix where the diagonal elements are $2, -3,$ and $5$?

If $\Delta = \begin{vmatrix} 0 & a \\ -a & 0 \end{vmatrix}$, find the value of $\Delta$.

Without expanding, evaluate: $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \end{vmatrix}$.

If $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = 10$, find the value of $\begin{vmatrix} a & b & c \\ g & h & i \\ d & e & f \end{vmatrix}$.

If $A$ is a square matrix of order 3 and $|A| = 7$, find the value of $|3A|$.

Without expanding, show that $\begin{vmatrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{vmatrix} = 0$. (Hint: Apply $C_3 \to C_3 - 9C_2$).

Evaluate without expanding: $\begin{vmatrix} x & y & z \\ a & b & c \\ x+a & y+b & z+c \end{vmatrix}$.

Let $A$ be a skew-symmetric matrix of odd order (e.g., $3 \times 3$). What is the value of $|A|$?

Given $|A| = -4$, find the determinant of $A^T$.

Use properties to evaluate $\begin{vmatrix} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \end{vmatrix}$.

If we apply the operation $R_1 \to R_1 + 2R_2$ on a determinant $\Delta$, what will be the new value of the determinant?

If every element of a $3 \times 3$ determinant is multiplied by $-1$, how does the value of the determinant change?

Find the area of the triangle with vertices $(2, 7)$, $(1, 1)$, and $(10, 8)$ using determinants.

Find the area of the triangle whose vertices are $(-2, -3)$, $(3, 2)$, and $(-1, -8)$.

Use determinants to show that the points $(a, b+c)$, $(b, c+a)$, and $(c, a+b)$ are collinear.

Find the equation of the line joining $(1, 2)$ and $(3, 6)$ using determinants.

If the points $(2, -3)$, $(k, -1)$, and $(0, 4)$ are collinear, find the value of $k$.

Find the value(s) of $k$ if the area of the triangle is 4 sq. units and vertices are $(k, 0)$, $(4, 0)$, and $(0, 2)$.

Find $k$ if the area of a triangle is $35$ sq. units with vertices $(2, -6)$, $(5, 4)$, and $(k, 4)$.

Can the area of a triangle obtained using the determinant formula be negative? If yes, how do we report the final area?

Find the area of the triangle with vertices $(0, 0)$, $(0, 5)$, and $(4, 0)$.

Using determinants, find the equation of the line joining $(3, 1)$ and $(9, 3)$.

Find the minor $M_{21}$ and cofactor $A_{21}$ of the element in the matrix $\begin{bmatrix} 1 & -2 \\ 4 & 3 \end{bmatrix}$.

Write the cofactor matrix for $A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$.

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

For the determinant $\Delta = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}$, find the minor $M_{22}$.

For the determinant in Q34, find the cofactor $A_{23}$.

If $A$ is a square matrix of order 3 and $|A| = 6$, find the value of $|adj(A)|$.

If $A$ is a non-singular matrix of order 2 and $|A| = 4$, find $A \cdot (adj A)$.

Verify that $A(adj A) = |A|I$ for the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. (Find $|A|$ first).

If the value of a determinant is $\Delta$, what is the sum of the products of elements of any row with their corresponding cofactors?

If $A = \begin{bmatrix} 5 & -2 \\ 3 & -2 \end{bmatrix}$, find $adj(A)$.

Find the inverse of the matrix $A = \begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}$.

Determine if the matrix $A = \begin{bmatrix} 6 & 4 \\ 3 & 2 \end{bmatrix}$ is invertible.

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

For what value of $x$ is the matrix $A = \begin{bmatrix} 3 - x & 2 \\ 4 & 1 \end{bmatrix}$ singular?

Write the system of equations $2x + 3y = 5$ and $x - 2y = 4$ in the matrix form $AX = B$.

Check the consistency of the system: $x + 2y = 2$, $2x + 3y = 3$.

Check the consistency of the system: $x + 3y = 5$, $2x + 6y = 8$.

If a system of equations $AX = B$ has $|A| \neq 0$, the system has a _______ solution.

Solve the system $5x + 2y = 4$, $7x + 3y = 5$ using the matrix method.

If $A^{-1} = \begin{bmatrix} 3 & -1 \\ -5/2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$, find the solution matrix $X$.