University Linear Algebra Test Questions: Classifying Special Matrices

📚 Quick Study Guide

• 🔢 Symmetric Matrix: A square matrix that is equal to its transpose, i.e., $A = A^T$. This means $a_{ij} = a_{ji}$ for all $i$ and $j$.
• ➗ Skew-Symmetric Matrix: A square matrix that is equal to the negative of its transpose, i.e., $A = -A^T$. This means $a_{ij} = -a_{ji}$ for all $i$ and $j$, and the diagonal elements are zero.
• 🆔 Identity Matrix: A square matrix with ones on the main diagonal and zeros elsewhere. Denoted by $I$, it satisfies $AI = IA = A$ for any matrix $A$ of compatible size.
• orthogonal Orthogonal Matrix: A square matrix whose columns and rows are orthogonal unit vectors (orthonormal vectors), i.e., $A^T A = A A^T = I$.
• ✨ Diagonal Matrix: A square matrix in which all the entries outside the main diagonal are zero.
• 📈 Triangular Matrix: A square matrix where all elements above (lower triangular) or below (upper triangular) the main diagonal are zero.
• 💫 Hermitian Matrix: A complex square matrix that is equal to its conjugate transpose, i.e., $A = A^H$, where $A^H$ denotes the conjugate transpose of $A$.

Practice Quiz

1. Question 1: Which of the following matrices is a symmetric matrix?
   1. A) $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
   2. B) $\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$
   3. C) $\begin{bmatrix} 1 & 2 \\ -2 & 4 \end{bmatrix}$
   4. D) $\begin{bmatrix} 0 & 2 \\ 0 & 0 \end{bmatrix}$
2. Question 2: Which of the following matrices is a skew-symmetric matrix?
   1. A) $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$
   2. B) $\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$
   3. C) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
   4. D) $\begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}$
3. Question 3: Which of the following matrices is an identity matrix?
   1. A) $\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$
   2. B) $\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$
   3. C) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
   4. D) $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
4. Question 4: Which of the following matrices is an orthogonal matrix?
   1. A) $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$
   2. B) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
   3. C) $\begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}$
   4. D) $\begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{bmatrix}$
5. Question 5: Which of the following matrices is a diagonal matrix?
   1. A) $\begin{bmatrix} 1 & 2 \\ 0 & 4 \end{bmatrix}$
   2. B) $\begin{bmatrix} 1 & 0 \\ 0 & 4 \end{bmatrix}$
   3. C) $\begin{bmatrix} 1 & 0 \\ 2 & 4 \end{bmatrix}$
   4. D) $\begin{bmatrix} 0 & 1 \\ 4 & 0 \end{bmatrix}$
6. Question 6: Which of the following matrices is an upper triangular matrix?
   1. A) $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
   2. B) $\begin{bmatrix} 1 & 2 \\ 0 & 4 \end{bmatrix}$
   3. C) $\begin{bmatrix} 1 & 0 \\ 3 & 4 \end{bmatrix}$
   4. D) $\begin{bmatrix} 0 & 2 \\ 0 & 4 \end{bmatrix}$
7. Question 7: Which of the following is a Hermitian matrix? (where '*' denotes the complex conjugate)
   1. A) $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$
   2. B) $\begin{bmatrix} 1 & i \\ -i & 1 \end{bmatrix}$
   3. C) $\begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix}$
   4. D) $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$