If I is a 2 × 2 identity matrix, find the determinant of the matrix.

Answer Details

The identity matrix is a special kind of square matrix where all the entries on the main diagonal are \(1\) and all other entries are \(0\). For a \(2 \times 2\) identity matrix, the matrix looks like this:

\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} \]

To find the determinant of a \(2 \times 2\) matrix \( A = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \), the formula is:

\[ \det(A) = ad - bc \]

For the identity matrix, \( a = 1 \), \( b = 0 \), \( c = 0 \), and \( d = 1 \). Plugging these values into the formula:

\[ \det(I) = (1)(1) - (0)(0) = 1 - 0 = 1 \]

This means the determinant of the \(2 \times 2\) identity matrix is 1.

The determinant being \(1\) is significant because it means the identity matrix does not change the size (area, in two dimensions) of vectors it multiplies, and it is always invertible.