The inverse of a 3×3 matrix is given by:
`A^-1=("adj"A)/(detA)`
"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.
Consider the matrix:
`((5,6,1),(0,3,-3),(4,-7,2))`
The cofactor of 6 is
`|(0,-3),(4,2)|=0+12=12`
The cofactor of -3 is
`|(5,6),(4,-7)|=-35-24=-59`
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 1^{st} column becomes the 1^{st} row; 2^{nd} column becomes 2^{nd} row, etc.
Find the inverse of the following by using the adjoint matrix method:
`A=((5,6,1),(0,3,-3),(4,-7,2))`
Step 1:
Replace elements with cofactors and apply + and -
`((+(-15),-(12),+(-12)),(-(19),+(6),-(-59)),(+(-21),-(-15),+(15)))`
`=((-15,-12,-12),(-19,6,59),(-21,15,15))`
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
`A^-1=("adj"A)/detA`
So
`A^-1=("adj"A)/detA`
`=1/-159((-15,-19,-21),(-12,6,15),(-12,59,15))``
`A^-1=((0.094,0.119,0.132),(0.075,-0.038,-0.094),(0.075,-0.371,-0.094))`
Find the inverse of
`((-2,6,1),(0,3,-3),(4,-7,3))`
using Method 2.
`text(C of) A` `=((+(-12),-(12),+(-12)),(-(25),+(-10),-(-10)),(+(-21),-(6),+(-6)))`
`=((-12,-12,-12),(-25,-10,10),(-21,-6,-6))`
Interchange rows and columns:
`"adj"A=((-12,-25,-21),(-12,-10,-6),(-12,10,-6))`
`"det"A`
`=|(-2,6,1),(0,3,-3),(4,-7,3)|`
`=2(9-21)+4(-21)`
`=-60`
So
`A^-1=("adj"A)/(detA)`
`=1/-60((-12,-25,-21),(-12,-10,-6),(-12,10,-6))`
`=( (1/5,5/12,7/20),(1/5,1/6,1/10),(1/5,-1/6,1/10))`
`=((0.2,0.417,0.35),(0.2,0.167,0.1),(0.2,-0.167,0.1))`
Now let's see how to do all this more appropriately using a computer...
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