Matrices - Row Operations for Finding Inverse of Matrix (3x3) (Level 1)

Math worksheet on 'Matrices - Row Operations for Finding Inverse of Matrix (3x3) (Level 1)'. Part of a broader unit on 'Matrices' Learn online: app.mobius.academy/math/units/matrices/

Multiply row 3 of the augmented matrix by whatever factor is needed to solve for that row of the inverse matrix

$A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 5 & 5 & \vert & 1 & 0 & 0 \\ 6 & 5 & 3 & \vert & 0 & 1 & 0 \\ 0 & 0 & 0.25 & \vert & 0 & 0 & 1 \end{array} } \right]$

a $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 5 & 5 & \vert & 1 & 0 & 0 \\ 6 & 5 & 3 & \vert & 0 & 1 & 0 \\ 0 & 0 & 0.06 & \vert & 0 & 0 & 6 \end{array} } \right]$

b $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 5 & 5 & \vert & 1 & 0 & 0 \\ 6 & 5 & 3 & \vert & 0 & 1 & 0 \\ 0 & 0 & 0.06 & \vert & 0 & 0 & -2 \end{array} } \right]$

c $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 5 & 5 & \vert & 1 & 0 & 0 \\ 6 & 5 & 3 & \vert & 0 & 1 & 0 \\ 0 & 0 & 0.06 & \vert & 0 & 0 & 5 \end{array} } \right]$

d $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 5 & 5 & \vert & 1 & 0 & 0 \\ 6 & 5 & 3 & \vert & 0 & 1 & 0 \\ 0 & 0 & 1 & \vert & 0 & 0 & 4 \end{array} } \right]$

e $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 5 & 5 & \vert & 1 & 0 & 0 \\ 6 & 5 & 3 & \vert & 0 & 1 & 0 \\ 0 & 0 & 0.06 & \vert & 0 & 0 & 3 \end{array} } \right]$

f $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 5 & 5 & \vert & 1 & 0 & 0 \\ 6 & 5 & 3 & \vert & 0 & 1 & 0 \\ 0 & 0 & 0.06 & \vert & 0 & 0 & 7 \end{array} } \right]$

Multiply row 2 of the augmented matrix by whatever factor is needed to solve for that row of the inverse matrix

$A LaTex expression showing \left[ {\begin{array} {cccc} 5 & 6 & 9 & \vert & 1 & 0 & 0 \\ 0 & 0.33 & 0 & \vert & 0 & 1 & 0 \\ 9 & 7 & 6 & \vert & 0 & 0 & 1 \end{array} } \right]$

a $A LaTex expression showing \left[ {\begin{array} {cccc} 5 & 6 & 9 & \vert & 1 & 0 & 0 \\ 0 & 0.11 & 0 & \vert & 0 & 2.25 & 0 \\ 9 & 7 & 6 & \vert & 0 & 0 & 1 \end{array} } \right]$

b $A LaTex expression showing \left[ {\begin{array} {cccc} 5 & 6 & 9 & \vert & 1 & 0 & 0 \\ 0 & 0.11 & 0 & \vert & 0 & 5.25 & 0 \\ 9 & 7 & 6 & \vert & 0 & 0 & 1 \end{array} } \right]$

c $A LaTex expression showing \left[ {\begin{array} {cccc} 5 & 6 & 9 & \vert & 1 & 0 & 0 \\ 0 & 0.11 & 0 & \vert & 0 & -2.25 & 0 \\ 9 & 7 & 6 & \vert & 0 & 0 & 1 \end{array} } \right]$

d $A LaTex expression showing \left[ {\begin{array} {cccc} 5 & 6 & 9 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 3 & 0 \\ 9 & 7 & 6 & \vert & 0 & 0 & 1 \end{array} } \right]$

e $A LaTex expression showing \left[ {\begin{array} {cccc} 5 & 6 & 9 & \vert & 1 & 0 & 0 \\ 0 & 0.11 & 0 & \vert & 0 & 3.75 & 0 \\ 9 & 7 & 6 & \vert & 0 & 0 & 1 \end{array} } \right]$

f $A LaTex expression showing \left[ {\begin{array} {cccc} 5 & 6 & 9 & \vert & 1 & 0 & 0 \\ 0 & 0.11 & 0 & \vert & 0 & 1.5 & 0 \\ 9 & 7 & 6 & \vert & 0 & 0 & 1 \end{array} } \right]$

Swap rows 1 and 2 of the augmented matrix to solve those rows of the inverse matrix

$A LaTex expression showing \left[ {\begin{array} {cccc} 0 & 1 & 0 & \vert & 1 & 0 & 0 \\ 1 & 0 & 0 & \vert & 0 & 1 & 0 \\ 7 & 7 & 4 & \vert & 0 & 0 & 1 \end{array} } \right]$

a $A LaTex expression showing \left[ {\begin{array} {cccc} 1 & 0 & 0 & \vert & 0 & 1 & 0 \\ 0 & 1 & 0 & \vert & 1 & 0 & 0 \\ 7 & 7 & 4 & \vert & 0 & 0 & 1 \end{array} } \right]$

b $A LaTex expression showing \left[ {\begin{array} {cccc} 1 & 0 & 0 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 1 & 0 \\ 7 & 7 & 4 & \vert & 0 & 0 & 1 \end{array} } \right]$

c $A LaTex expression showing \left[ {\begin{array} {cccc} 1 & 0 & 0 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 1 & 0 & 0 \\ 7 & 7 & 4 & \vert & 0 & 0 & 1 \end{array} } \right]$

d $A LaTex expression showing \left[ {\begin{array} {cccc} 1 & 0 & 0 & \vert & 0 & 0 & 1 \\ 0 & 1 & 0 & \vert & 0 & 1 & 0 \\ 7 & 7 & 4 & \vert & 0 & 0 & 1 \end{array} } \right]$

e $A LaTex expression showing \left[ {\begin{array} {cccc} 1 & 0 & 0 & \vert & 0 & 1 & 0 \\ 0 & 1 & 0 & \vert & 0 & 1 & 0 \\ 7 & 7 & 4 & \vert & 0 & 0 & 1 \end{array} } \right]$

f $A LaTex expression showing \left[ {\begin{array} {cccc} 1 & 0 & 0 & \vert & 0 & 0 & 1 \\ 0 & 1 & 0 & \vert & 0 & 0 & 1 \\ 7 & 7 & 4 & \vert & 0 & 0 & 1 \end{array} } \right]$

Divide row 3 of the augmented matrix by whatever factor is needed to solve for that row of the inverse matrix

$A LaTex expression showing \left[ {\begin{array} {cccc} 3 & 7 & 2 & \vert & 1 & 0 & 0 \\ 8 & 8 & 0 & \vert & 0 & 1 & 0 \\ 0 & 0 & 4 & \vert & 0 & 0 & 1 \end{array} } \right]$

a $A LaTex expression showing \left[ {\begin{array} {cccc} 3 & 7 & 2 & \vert & 1 & 0 & 0 \\ 8 & 8 & 0 & \vert & 0 & 1 & 0 \\ 0 & 0 & 16 & \vert & 0 & 0 & -6 \end{array} } \right]$

b $A LaTex expression showing \left[ {\begin{array} {cccc} 3 & 7 & 2 & \vert & 1 & 0 & 0 \\ 8 & 8 & 0 & \vert & 0 & 1 & 0 \\ 0 & 0 & 16 & \vert & 0 & 0 & 3 \end{array} } \right]$

c $A LaTex expression showing \left[ {\begin{array} {cccc} 3 & 7 & 2 & \vert & 1 & 0 & 0 \\ 8 & 8 & 0 & \vert & 0 & 1 & 0 \\ 0 & 0 & 1 & \vert & 0 & 0 & 0.25 \end{array} } \right]$

d $A LaTex expression showing \left[ {\begin{array} {cccc} 3 & 7 & 2 & \vert & 1 & 0 & 0 \\ 8 & 8 & 0 & \vert & 0 & 1 & 0 \\ 0 & 0 & 16 & \vert & 0 & 0 & 6 \end{array} } \right]$

e $A LaTex expression showing \left[ {\begin{array} {cccc} 3 & 7 & 2 & \vert & 1 & 0 & 0 \\ 8 & 8 & 0 & \vert & 0 & 1 & 0 \\ 0 & 0 & 16 & \vert & 0 & 0 & -3 \end{array} } \right]$

f $A LaTex expression showing \left[ {\begin{array} {cccc} 3 & 7 & 2 & \vert & 1 & 0 & 0 \\ 8 & 8 & 0 & \vert & 0 & 1 & 0 \\ 0 & 0 & 16 & \vert & 0 & 0 & -4 \end{array} } \right]$

Add row 1 to row 2 of the augmented matrix as many times as needed to solve for row 2 of the inverse matrix

$A LaTex expression showing \left[ {\begin{array} {cccc} 4 & 0 & 0 & \vert & 1 & 0 & 0 \\ -8 & 1 & 0 & \vert & 0 & 1 & 0 \\ 1 & 7 & 1 & \vert & 0 & 0 & 1 \end{array} } \right]$

a $A LaTex expression showing \left[ {\begin{array} {cccc} 4 & 0 & 0 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & -3 & 1 & 0 \\ 1 & 7 & 1 & \vert & 0 & 0 & 1 \end{array} } \right]$

b $A LaTex expression showing \left[ {\begin{array} {cccc} 4 & 0 & 0 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 1 & 1 & 0 \\ 1 & 7 & 1 & \vert & 0 & 0 & 1 \end{array} } \right]$

c $A LaTex expression showing \left[ {\begin{array} {cccc} 4 & 0 & 0 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & -6 & 1 & 0 \\ 1 & 7 & 1 & \vert & 0 & 0 & 1 \end{array} } \right]$

d $A LaTex expression showing \left[ {\begin{array} {cccc} 4 & 0 & 0 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 6 & 1 & 0 \\ 1 & 7 & 1 & \vert & 0 & 0 & 1 \end{array} } \right]$

e $A LaTex expression showing \left[ {\begin{array} {cccc} 4 & 0 & 0 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 2 & 1 & 0 \\ 1 & 7 & 1 & \vert & 0 & 0 & 1 \end{array} } \right]$

f $A LaTex expression showing \left[ {\begin{array} {cccc} 4 & 0 & 0 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 4 & 1 & 0 \\ 1 & 7 & 1 & \vert & 0 & 0 & 1 \end{array} } \right]$

Add row 3 to row 2 of the augmented matrix as many times as needed to solve for row 2 of the inverse matrix

$A LaTex expression showing \left[ {\begin{array} {cccc} 9 & 3 & 2 & \vert & 1 & 0 & 0 \\ 0 & 1 & -9 & \vert & 0 & 1 & 0 \\ 0 & 0 & 3 & \vert & 0 & 0 & 1 \end{array} } \right]$

a $A LaTex expression showing \left[ {\begin{array} {cccc} 9 & 3 & 2 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 1 & -6 \\ 0 & 0 & 3 & \vert & 0 & 0 & 1 \end{array} } \right]$

b $A LaTex expression showing \left[ {\begin{array} {cccc} 9 & 3 & 2 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 1 & 9 \\ 0 & 0 & 3 & \vert & 0 & 0 & 1 \end{array} } \right]$

c $A LaTex expression showing \left[ {\begin{array} {cccc} 9 & 3 & 2 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 1 & -3 \\ 0 & 0 & 3 & \vert & 0 & 0 & 1 \end{array} } \right]$

d $A LaTex expression showing \left[ {\begin{array} {cccc} 9 & 3 & 2 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 1 & -4.5 \\ 0 & 0 & 3 & \vert & 0 & 0 & 1 \end{array} } \right]$

e $A LaTex expression showing \left[ {\begin{array} {cccc} 9 & 3 & 2 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 1 & 10.5 \\ 0 & 0 & 3 & \vert & 0 & 0 & 1 \end{array} } \right]$

f $A LaTex expression showing \left[ {\begin{array} {cccc} 9 & 3 & 2 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 1 & 3 \\ 0 & 0 & 3 & \vert & 0 & 0 & 1 \end{array} } \right]$

Swap rows 2 and 3 of the augmented matrix to solve those rows of the inverse matrix

$A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 1 & 5 & \vert & 1 & 0 & 0 \\ 0 & 0 & 1 & \vert & 0 & 1 & 0 \\ 0 & 1 & 0 & \vert & 0 & 0 & 1 \end{array} } \right]$

a $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 1 & 5 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 0 & 1 \\ 0 & 0 & 1 & \vert & 0 & 0 & 1 \end{array} } \right]$

b $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 1 & 5 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 0 & 1 \\ 0 & 0 & 1 & \vert & 0 & 1 & 0 \end{array} } \right]$

c $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 1 & 5 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 0 & 1 \\ 0 & 0 & 1 & \vert & 1 & 0 & 0 \end{array} } \right]$

d $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 1 & 5 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 1 & 0 & 0 \\ 0 & 0 & 1 & \vert & 0 & 1 & 0 \end{array} } \right]$

e $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 1 & 5 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 1 & 0 \\ 0 & 0 & 1 & \vert & 1 & 0 & 0 \end{array} } \right]$

f $A LaTex expression showing \left[ {\begin{array} {cccc} 7 & 1 & 5 & \vert & 1 & 0 & 0 \\ 0 & 1 & 0 & \vert & 0 & 1 & 0 \\ 0 & 0 & 1 & \vert & 0 & 0 & 1 \end{array} } \right]$

Practice:

Row Operations for Finding Inverse of Matrix (3x3)

Matrices - Row Operations for Finding Inverse of Matrix (3x3) Worksheet