Inverse of a matrix by Gauss-Jordan elimination

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Inverse of a matrix by Gauss-Jordan elimination

Linear Algebra - Matrices: (lesson 3 of 3)

Inverse of a matrix by Gauss-Jordan elimination

To find the inverse of matrix $A$ , using Gauss-Jordan elimination, it must be found the sequence of elementary row operations that reduces $A$ to the identity and, then, the same operations on $I_n$ must be performed to obtain $A^{-1}$ .

Inverse of 2 $\times$ 2 matrices

Example 1: Find the inverse of

$A = \left[ {} \right]$

Solution:

Step 1: Adjoin the identity matrix to the right side of $A$ :

$A = \left[ {\left| {} \right.} \right]$

Step 2: Apply row operations to this matrix until the left side is reduced to $I$ . The computations are:

Step 3: Conclusion: The inverse matrix is:

$A^{-1} = \left[ {} \right]$

Not invertible matrix

If $A$ is not invertible, then, a zero row will show up on the left side.

Example 2: Find the inverse of

$A = \left[ {} \right]$

Solution:

Step 1: Adjoin the identity matrix to the right side of A:

$\left[ {\left| {} \right.} \right]$

Step 2: Apply row operations

$\left[ {\left| {} \right.} \right]Row2 = Row2 + \color{red}{2} \cdot \color{blue}{Row1}$

$\left[ {\left| {} \right.} \right]_{\color{red}{\leftarrow ZERO \ \ ROW}}$

Step 3: Conclusion: This matrix is not invertible.

Inverse of 3 $\times$ 3 matrices

Example 1: Find the inverse of

$A = \left[ {} \right]$

Solution:

Step 1: Adjoin the identity matrix to the right side of A:

$\left[ {\left| {} \right.} \right]$

Step 2: Apply row operations to this matrix until the left side is reduced to I. The computations are:

$\left[ {\left| {} \right.} \right]\mathop { - - - - - - - \to }\limits_{R3 = R3 \color{red}{-} R1}^{R2 = R2 - \color{blue}{2} \cdot R1} \left[ {\left| {} \right.} \right]$

$\left[ {\left| {} \right.} \right]\mathop { - - - - - - - \to }\limits_{R3 = R3 \color{blue}{+ 2} \cdot R2}^{} \left[ {\left| {} \right.} \right]$

$\left[ {\left| {} \right.} \right]\mathop { - - - - - - - \to }\limits_{R3 = - 1 \cdot R3}^{} \left[ {\left| {} \right.} \right]$

$\left[ {\left| {} \right.} \right]\mathop { - - - - - - - \to }\limits_{R2 = R2 + 3 \cdot R3}^{R1 = R1 - 3 \cdot R3} \left[ {\left| {} \right.} \right]$

$\left[ {\left| {} \right.} \right]\mathop { - - - - - - - \to }\limits_{}^{R1 = R1 - 2 \cdot R2} \left[ {\left| {} \right.} \right]$

Step 3: Conclusion: The inverse matrix is:

$A^{ - 1} = \left[ {} \right]$

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