Properties of Determinants Exercise
Question # 1 (Covering: The determinant of an identity matrix is always 1.)
\[
\textbf{Question: Calculate the determinant of the identity matrix } I \textbf{ of order } 3 \textbf{ (3×3 matrix).}
\]
\[
I = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]
\[
\textbf{2. Determinant of a Triangular Matrix:}
\]
\[
\text{The determinant of a triangular matrix (upper or lower) is the product of its diagonal elements.}
\]
\[
\textbf{3. Row or Column with All Zeros:}
\]
\[
\text{If any row or column of a matrix consists entirely of zeros, the determinant of the matrix is zero.}
\]
\[
\textbf{4. Interchanging Rows or Columns:}
\]
\[
\text{If two rows or two columns of a matrix are swapped, the determinant of the matrix changes its sign.}
\]
\[
\textbf{5. Proportional Rows or Columns:}
\]
\[
\text{If two rows or columns of a matrix are proportional (one is a scalar multiple of the other), the determinant of the matrix is zero.}
\]
\[
\textbf{6. Addition of Rows or Columns:}
\]
\[
\text{Adding a scalar multiple of one row (or column) to another row (or column) does not change the determinant.}
\]
\[
\textbf{7. Scalar Multiplication:}
\]
\[
\text{If a row or column of a matrix is multiplied by a scalar } k \text{, the determinant of the matrix is also multiplied by } k.
\]
\[
\textbf{8. Determinant of a Singular Matrix:}
\]
\[
\text{If the determinant of a matrix is zero, the matrix is singular and does not have an inverse.}
\]
\[
\textbf{9. Determinant of the Product of Matrices:}
\]
\[
\text{For two matrices } A \text{ and } B, \det(AB) = \det(A) \cdot \det(B).
\]
\[
\textbf{10. Determinant of the Transpose:}
\]
\[
\text{The determinant of a matrix is equal to the determinant of its transpose: } \det(A) = \det(A^T).
\]
\[
\textbf{11. Determinant of Diagonal Matrices:}
\]
\[
\text{The determinant of a diagonal matrix is the product of its diagonal elements.}
\]
\[
\textbf{12. Determinant of Block Matrices:}
\]
\[
\text{If } A \text{ is a block diagonal matrix, its determinant is the product of the determinants of its diagonal blocks.}
\]
\[
\textbf{13. Cofactor Expansion (Laplace Expansion):}
\]
\[
\text{The determinant of a matrix can be computed using cofactor expansion along any row or column.}
\]
\[
\textbf{14. Determinant of an Inverse Matrix:}
\]
\[
\text{If } A \text{ is invertible, } \det(A^{-1}) = \frac{1}{\det(A)}.
\]
\[
\textbf{15. Determinant of Scalar Multiplication:}
\]
\[
\text{If a matrix } A \text{ is multiplied by a scalar } k, \det(kA) = k^n \cdot \det(A) \text{, where } n \text{ is the order of the matrix.}
\]
\[
\textbf{16. Row or Column Repetition:}
\]
\[
\text{If any two rows or columns of a matrix are identical, the determinant of the matrix is zero.}
\]