1. **Define a matrix and write its general form.** A matrix is a rectangular array of numbers arranged in rows and columns. The general form of a matrix with $m$ rows and $n$ columns is: $$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$ 2. **What is the order of the matrix** $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} $$ The order of a matrix is given by the number of rows by the number of columns. Here, $A$ has 2 rows and 3 columns, so the order is $2 \times 3$. 3. **Define a row matrix with an example.** A row matrix is a matrix with only one row. Example: $$ R = \begin{bmatrix} 4 & 7 & 1 \end{bmatrix} $$ 4. **Define a column matrix with an example.** A column matrix is a matrix with only one column. Example: $$ C = \begin{bmatrix} 3 \\ 5 \\ 2 \end{bmatrix} $$ 5. **What is a square matrix?** A square matrix is a matrix with the same number of rows and columns, i.e., order $n \times n$. 6. **Write the zero matrix of order 3 × 3.** The zero matrix has all elements zero: $$ O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$ 7. **Define the identity matrix of order 3.** The identity matrix has 1s on the main diagonal and 0s elsewhere: $$ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$ 8. **State the condition for addition of two matrices.** Two matrices can be added only if they have the same order. 9. **Find $A + B$, if** $$ A = \begin{bmatrix} 2 & 3 & 1 \\ 0 & 1 & 2 \\ 4 & 0 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 & 2 \\ 3 & 4 & 1 \\ 0 & 5 & 2 \end{bmatrix} $$ Add corresponding elements: $$ A + B = \begin{bmatrix} 2+1 & 3+0 & 1+2 \\ 0+3 & 1+4 & 2+1 \\ 4+0 & 0+5 & 3+2 \end{bmatrix} = \begin{bmatrix} 3 & 3 & 3 \\ 3 & 5 & 3 \\ 4 & 5 & 5 \end{bmatrix} $$ 10. **Find $A^T$, if** $$ A = \begin{bmatrix} 2 & 3 & 1 \\ 5 & 1 & 4 \\ 0 & 2 & 6 \end{bmatrix} $$ The transpose $A^T$ is obtained by swapping rows and columns: $$ A^T = \begin{bmatrix} 2 & 5 & 0 \\ 3 & 1 & 2 \\ 1 & 4 & 6 \end{bmatrix} $$ 11. **Define symmetric and skew-symmetric matrices with examples.** - Symmetric matrix: $A = A^T$. Example: $$ \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} $$ - Skew-symmetric matrix: $A^T = -A$ and diagonal elements are zero. Example: $$ \begin{bmatrix} 0 & 4 \\ -4 & 0 \end{bmatrix} $$ 12. **If** $$ A = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 0 & 4 \\ 2 & 3 & 5 \end{bmatrix} $$ **find $|A|$ (determinant).** Calculate determinant by expansion: $$ |A| = 3 \begin{vmatrix} 0 & 4 \\ 3 & 5 \end{vmatrix} - 2 \begin{vmatrix} 1 & 4 \\ 2 & 5 \end{vmatrix} + 1 \begin{vmatrix} 1 & 0 \\ 2 & 3 \end{vmatrix} $$ Calculate minors: $$ = 3(0 \times 5 - 4 \times 3) - 2(1 \times 5 - 4 \times 2) + 1(1 \times 3 - 0 \times 2) $$ $$ = 3(0 - 12) - 2(5 - 8) + 1(3 - 0) $$ $$ = 3(-12) - 2(-3) + 3 = -36 + 6 + 3 = -27 $$ 13. **What is the determinant of an identity matrix of order 3?** The determinant of the identity matrix is always 1. 14. **Define the inverse of a matrix.** The inverse of a matrix $A$ is a matrix $A^{-1}$ such that: $$ A A^{-1} = A^{-1} A = I $$ where $I$ is the identity matrix. 15. **State the condition for a matrix to be invertible.** A matrix is invertible if and only if its determinant is non-zero. 16. **Define the rank of a matrix.** The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. 17. **Explain what is meant by a diagonal matrix and give an example.** A diagonal matrix is a square matrix where all off-diagonal elements are zero. Example: $$ D = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 7 \end{bmatrix} $$