Understanding Square Matrices: Definition, Properties, and Examples

📚 Quick Study Guide

• 📏 Definition: A square matrix is a matrix with an equal number of rows and columns (i.e., an $n \times n$ matrix).
• ➕ Addition & Subtraction: Two square matrices of the same size can be added or subtracted element-wise.
• ✖️ Multiplication: Square matrices can be multiplied, provided the dimensions are compatible (inner dimensions must match).
• 🆔 Identity Matrix: An identity matrix ($I_n$) is a square matrix with 1s on the main diagonal and 0s elsewhere. When multiplied by any matrix $A$ of compatible dimensions, it leaves $A$ unchanged: $AI = IA = A$.
• 🔄 Transpose: The transpose of a square matrix $A$ (denoted as $A^T$) is obtained by interchanging its rows and columns.
• 📉 Determinant: Every square matrix has a determinant, which is a scalar value that can be computed from its elements. It indicates if the matrix is invertible.
• ➗ Inverse: A square matrix $A$ is invertible if there exists a matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$. Only square matrices can have inverses.

Practice Quiz

1. Which of the following matrices is a square matrix?
   1. $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
   2. $B = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$
   3. $C = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$
   4. $D = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}$
2. What is the size of a square matrix with 5 rows?
   1. $5 \times 6$
   2. $6 \times 5$
   3. $5 \times 5$
   4. $1 \times 5$
3. Which of the following is a characteristic of the identity matrix?
   1. All elements are 1.
   2. All elements are 0.
   3. It is a rectangular matrix.
   4. It has 1s on the main diagonal and 0s elsewhere.
4. If $A$ is a square matrix and $A^T$ is its transpose, what operation is performed to obtain $A^T$ from $A$?
   1. Adding rows and columns
   2. Subtracting rows from columns
   3. Interchanging rows and columns
   4. Multiplying rows and columns
5. Which of the following is necessary for a square matrix to have an inverse?
   1. Its determinant must be equal to 1.
   2. Its determinant must be non-zero.
   3. Its determinant must be equal to zero.
   4. It must be an identity matrix.
6. Given $A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$, what is $A^T$ (the transpose of A)?
   1. $\begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix}$
   2. $\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$
   3. $\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix}$
   4. $\begin{bmatrix} 3 & 4 \\ 2 & 1 \end{bmatrix}$
7. Which of the following operations is ALWAYS possible with two square matrices of the SAME size?
   1. Multiplication
   2. Finding the inverse
   3. Addition
   4. Division