Matrices: Identity and Inverse

Identity Matrix

• Denoted as matrix I.
• Multiplicative identity.
• For a 2x2 matrix: $\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$
• For a 3x3 matrix: $\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$
• Always has ones down the main diagonal and zeros everywhere else.

Inverse Matrix

• If $A * B = I$, then A and B are inverses of each other.
• Notation: $A^{-1}$

Finding the Inverse of a 2x2 Matrix

1. Calculate the Determinant:

   • For a matrix $\begin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is calculated as: $det(A) = ad - bc$

2. Inverse Formula:
   $A^{-1} = \frac{1}{det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$

   • Switch the positions of $a$ and $d$.
   • Take the negative of $b$ and $c$.
   • Multiply the resulting matrix by one over the determinant.

Example

Given matrix A:

$\begin{bmatrix} 3 & 7 \ -2 & 4 \end{bmatrix}$

1. Calculate the Determinant:

   $det(A) = (3 * 4) - (7 * -2) = 12 - (-14) = 26$

2. Apply the Inverse Formula:

   $A^{-1} = \frac{1}{26} \begin{bmatrix} 4 & -7 \ 2 & 3 \end{bmatrix} = \begin{bmatrix} \frac{4}{26} & \frac{-7}{26} \ \frac{2}{26} & \frac{3}{26} \end{bmatrix} = \begin{bmatrix} \frac{2}{13} & \frac{-7}{26} \ \frac{1}{13} & \frac{3}{26} \end{bmatrix}$

Important Note

• If the determinant of a matrix is zero, the matrix does not have an inverse because division by zero is undefined.

Calculator Usage

• Calculators can quickly compute the inverse of matrices, including 3x3 and larger matrices.
• The notation for the inverse matrix on a calculator is typically raising the matrix to the power of -1 (e.g., $A^{-1}$).
• Calculators can also calculate the determinant of matrices.

Example of a 3x3 Matrix Inverse

• Original Matrix:

  $\begin{bmatrix} 1 & 2 & 3 \ -7 & 8 & 9 \ 0 & 4 & 5 \end{bmatrix}$

• Calculator provides the inverse directly.

• Determinant of the original matrix is -10.