Linear Algebra Quiz: Determinants, Invertibility, and det(A) ≠ 0

📚 Quick Study Guide

• ➕ The determinant of a matrix, denoted as $det(A)$ or $|A|$, is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix.
• 🔢 For a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is calculated as $det(A) = ad - bc$.
• ➗ For a 3x3 matrix, the determinant can be found using cofactor expansion.
• 🔑 A square matrix $A$ is invertible (i.e., has an inverse $A^{-1}$) if and only if its determinant is non-zero, i.e., $det(A) ≠ 0$.
• 🔄 If $det(A) ≠ 0$, the inverse $A^{-1}$ exists, and multiplying $A$ by its inverse results in the identity matrix $I$: $A A^{-1} = A^{-1} A = I$.
• 💡 If $det(A) = 0$, the matrix $A$ is singular (non-invertible). Its columns (or rows) are linearly dependent.
• 📐 The determinant is used to find the area (in 2D) or volume (in 3D) scaling factor of a linear transformation.

Practice Quiz

1. What is the determinant of the matrix $A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$?

   1. 11
   2. 5
   3. -2
   4. 2

2. If $det(A) = 0$, what can be concluded about matrix $A$?

   1. $A$ is invertible.
   2. $A$ is singular (non-invertible).
   3. $A$ is an identity matrix.
   4. $A$ is orthogonal.

3. Which of the following statements is true regarding an invertible matrix $A$?

   1. $det(A) = 0$
   2. $det(A) ≠ 0$
   3. $det(A) = 1$
   4. $det(A)$ is undefined.

4. What is the determinant of the identity matrix $I_n$ of size $n \times n$?

   1. 0
   2. 1
   3. $n$
   4. -1

5. For what value of $k$ is the matrix $A = \begin{bmatrix} 1 & 2 \\ 2 & k \end{bmatrix}$ singular?

   1. $k = 1$
   2. $k = 2$
   3. $k = 3$
   4. $k = 4$

6. If $A$ and $B$ are square matrices of the same size, and $det(A) = 2$ and $det(B) = 3$, what is $det(AB)$?

   1. 5
   2. 6
   3. -1
   4. 0

7. If $det(A) = -1$, what is the determinant of $A^{-1}$?

   1. -1
   2. 1
   3. 0
   4. Undefined