1.What is the determinant of a $1 \times 1$ matrix $A = [-7]$?
2.Evaluate the determinant: $\begin{vmatrix} 2 & 4 \\ 1 & 3 \end{vmatrix}$.
3.If $\begin{vmatrix} x & 2 \\ 3 & 4 \end{vmatrix} = 10$, find the value of $x$.
4.Evaluate the determinant: $\begin{vmatrix} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{vmatrix}$.
5.Evaluate: $\begin{vmatrix} x & x+1 \\ x-1 & x \end{vmatrix}$.
6.What is the value of the determinant of an Identity Matrix of order $3$ ($I_3$)?
7.If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$, find $|A|$.
8.Is the determinant defined for a non-square matrix? (Yes/No)
9.Expand and evaluate: $\begin{vmatrix} 1 & 2 & 0 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{vmatrix}$.
10.Find the value of $x$ if $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$.
11.If the rows and columns of a determinant are interchanged, how does the value of the determinant change?
12.What is the relationship between $|A|$ and $|A^T|$ (determinant of the transpose of $A$)?
13.If any two rows (or columns) of a determinant are identical, what is the value of the determinant?
14.Evaluate without expanding: $\begin{vmatrix} 2 & 3 & 4 \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{vmatrix}$.
15.If $A$ is a square matrix of order $3$ and $|A| = 4$, what is the value of $|2A|$?
16.Fill in the blank: For a square matrix $A$ of order $n$, $|kA| = $ _______ $|A|$.
17.What happens to the sign of a determinant if any two adjacent rows are interchanged?
18.If every element of a row in a determinant is multiplied by a constant $k$, the value of the new determinant is multiplied by _______.
19.True or False: $|A + B| = |A| + |B|$ for all square matrices $A$ and $B$.
20.Evaluate $\begin{vmatrix} 1 & 2 & 3 \\ 10 & 20 & 30 \\ 5 & 7 & 9 \end{vmatrix}$ using properties.
21.Write the determinant formula for the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$.
22.If three points are collinear, what is the area of the triangle formed by them?
23.Find the area of the triangle with vertices $(1, 0)$, $(6, 0)$, and $(4, 3)$.
24.If the area of a triangle is $35$ sq units with vertices $(2, -6)$, $(5, 4)$ and $(k, 4)$, what equation would you set up to find $k$?
25.What is the "Minor" ($M_{ij}$) of an element $a_{ij}$ in a determinant?
26.Find the minor $M_{11}$ of the element $2$ in the determinant $\begin{vmatrix} 2 & -3 \\ 4 & 5 \end{vmatrix}$.
27.How is the Cofactor ($A_{ij}$ or $C_{ij}$) related to the Minor ($M_{ij}$)? (Write the formula).
28.Find the cofactor $A_{12}$ for the element $-3$ in the determinant $\begin{vmatrix} 2 & -3 \\ 4 & 5 \end{vmatrix}$.
29.If elements of a row are multiplied by the cofactors of any *other* row, what is the sum of their products?
30.If elements of a row are multiplied by their *own* corresponding cofactors, what is the sum equal to?
31.The Adjoint of a square matrix $A$, denoted by $adj(A)$, is defined as the transpose of which matrix?
32.Write the direct trick/rule to find the adjoint of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$.
33.State the fundamental theorem connecting a matrix $A$, its adjoint, and its determinant: $A(adj A) = $ _______.
34.A square matrix $A$ is said to be "singular" if its determinant $|A|$ equals what value?
35.Write the formula to find the Inverse of a matrix $A$ ($A^{-1}$) using its adjoint.
36.Can you find the inverse of a singular matrix? (Yes/No)
37.If $A$ is a non-singular square matrix of order $n$, what is the formula for $|adj(A)|$ in terms of $|A|$?
38.If $A$ is a $3 \times 3$ matrix and $|A| = 5$, find the value of $|adj(A)|$.
39.True or False: $(AB)^{-1} = A^{-1}B^{-1}$.
40.What is the inverse of the Identity matrix $I$?
41.A system of linear equations is said to be "consistent" if it has at least one _______.
42.In the matrix equation $AX = B$, if $|A| \neq 0$, does the system have a unique solution or infinitely many solutions?
43.If $AX = B$ and $|A| = 0$, the system could either be inconsistent or have _______ solutions.
44.In Cramer's Rule for a system of 3 variables, $x = \frac{\Delta_1}{\Delta}$. For the system to have a unique solution, what must be true about $\Delta$?
45.A homogeneous system of linear equations ($AX = O$) is always consistent because it trivially has which solution?
46.For a homogeneous system $AX = O$, if $|A| = 0$, what kind of solutions does the system possess?
47.True or False: If a system of equations represents two parallel lines, the system is inconsistent.
48.When differentiating a $2 \times 2$ determinant where elements are functions of $x$, how many separate determinants are added in the resulting expression?
49.Let $\Delta(x) = \begin{vmatrix} f(x) & g(x) \\ a & b \end{vmatrix}$ where $a, b$ are constants. Write the expression for $\frac{d}{dx}\Delta(x)$.
50.If $\Delta = \begin{vmatrix} 1 & x \\ 0 & 1 \end{vmatrix}$, find the derivative of $\Delta$ with respect to $x$.
