Step 1: Write down the matrix for which we want to find the determinant: \( A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \).
Step 2: Understand that the determinant can be calculated using a formula or by checking if the rows are linearly dependent.
Step 3: Check if the rows of the matrix are linearly dependent. This means that one row can be expressed as a combination of the others.
Step 4: Notice that if you take the first row (1, 2, 3) and add it to the second row (4, 5, 6), you can see that the third row (7, 8, 9) is a linear combination of the first two rows.
Step 5: Since the rows are linearly dependent, the determinant of the matrix is 0.
Determinants – The determinant of a matrix is a scalar value that can be computed from its elements and provides important properties about the matrix, such as whether it is invertible.
Linear Dependence – Rows (or columns) of a matrix are linearly dependent if at least one row (or column) can be expressed as a linear combination of others, which results in a determinant of zero.
