Method 2 (an example of dinosaur mathematics - should be extinct)

The inverse of a 3×3 matrix is given by:


"adj A" is short for "the adjoint of A". We use cofactors (that we met earlier) to determine the adjoint of a matrix.


Recall: The cofactor of an element in a matrix is the value obtained by evaluating the determinant formed by the elements not in that particular row or column.

Example 2a

Consider the matrix:


The cofactor of 6 is


The cofactor of -3 is


We find the adjoint matrix by replacing each element in the matrix with its cofactor and applying a + or - sign as follows:


and then finding the transpose of the resulting matrix. The transpose means the 1st column becomes the 1st row; 2nd column becomes 2nd row, etc.

Example 2b

Find the inverse of the following by using the adjoint matrix method:



Step 1:

Replace elements with cofactors and apply + and -




Step 2

Transpose the matrix:

`"adj"A = ((-15,-19,-21),(-12,6,15),(-12,59,15))`

Before we can find the inverse of matrix A, we need det A:

`|(5,6,1),(0,3,-3),(4,-7,2)|` `=5(-15)+4(-21)` `=-159`

Now we have what we need to apply the formula






Example 2c

Find the inverse of


using Method 2.


`text(C of) A` `=((+(-12),-(12),+(-12)),(-(25),+(-10),-(-10)),(+(-21),-(6),+(-6)))`


Interchange rows and columns:









`=( (1/5,5/12,7/20),(1/5,1/6,1/10),(1/5,-1/6,1/10))`


Now let's see how to do all this more appropriately using a computer...

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