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How do you find the inverse of #A=##((-2, -2, -2), (-3, 2, -3), (1, -2, 1))#?

Question

How do you find the inverse of #A=##((-2, -2, -2), (-3, 2, -3), (1, -2, 1))#?
How do you find the inverse of #A=##((-2, -2, -2), (-3, 2, -3), (1, -2, 1))#?

Answers

This matrix is not invertible

Explanation:

For a matrix #A# to be invertible, the determinant #detA!=0#

Let's calculate

#detA=|(-2,-2,-2),(-3,2,-3),(1,-2,1)|#

#=-2*|(2,-3),(-2,1)|+2*|(-3,-3),(1,1)|-2*|(-3,2),(1,-2)|#

#=-2*(2-6)+2*(-3+3)-2*(6-2)#

#=8+0-8#

#=0#

As #detA=0#, matrix #A# is not invertible

The first and third column vectors are the same, ie linearly dependent. So by inspection the matrix is non invertible. In simple algebra terms, you have three unknowns and 2 equations.

If the column vectors are linearly dependent, it follows that the row vectors are too.

Here, you will find that the second row vector minus twice the first row vector gives you the third row vector.

These and that fact of the zero determinant and non-invertibility are all equivalent statements.


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