Complex Matrix Inverse

Introduction

In algebra, the determinant is a particular number related with some square matrix. The essential geometric significance of a determinant is a scale factor for compute when the matrix is considered as a linear transformation. Thus a 3× 3 matrix with determinant 3 when useful to a place of points with fixed area will convert those points into a place with double the area. Determinants are significant together in calculus, where they go into the replacement rule for various variables, and in multi linear algebra.

`A = [[A,B],[C,D]] If AD - BD != 0` , then A has an inverse, Denoted` A^-1`

`A^-1 = 1/(AD-BC) [[D,-B],[-C,A]]`

Complex Matrix Inverse - Examples:

Complex matrix inverse - Example 1:

Find the Complex inverse matrix of the matrix below?

`A = [[4,2],[3, 8]]`

Solution:

Inverse A =` 1/((8*4)-(3*2)) [[8,-3],[-2,4]]`

= `1/(32-6) [[8,-3],[-2,4]]`

=` 1/26[[8,-3],[-2,4]]`

= `1/26 * [[8,-3],[-2,4]] = [[8/26, -3/26],[-2/26,4/26]]`

= `[[4/13, -3/26], [-1/13, 2/13]]`

Complex matrix inverse - Example 2:

What is the inverse of the matrix below?

`[[9, 1], [6, 3]]`

Solution:

Inverse of the matrix is the

Inverse A =` 1/((9*3)-(6*1)) [[3,-1],[-6,9]]`

= `1/(27-6) [[3,-1],[-6,9]]`

= `1/21[[3,-1],[-6,9]]`

= `1/21 * [[3,-1],[-6,9]] = [[3/21, -1/21],[-6/21,9/21]]`

` [[3/21, -1/21],[-6/21,9/21]] = [[1/7, -1/21],[-2/7,3/7]]`

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Complex matrix inverse - Example 3:

What is the inverse of the matrix below?

A =  ` [[7, 1], [5, 3]]`

Solution:

Inverse of the matrix is the

Inverse A =` 1/((7*3)-(5*1)) [[3,-1],[-5,7]]`

= `1/(21-5) [[3,-1],[-5,7]]`

= `1/16[[3,-1],[-5,7]]`

= `1/16 * [[3,-1],[-5,7]]`

= `[[3/16, -1/16],[-6/16,9/16]]`

`[[3/16, -1/16],[-6/16,9/16]] = [[3/16, -1/16],[-3/8,9/16]]`

Complex Matrix Inverse - more Examples:

Complex matrix inverse - Example 1:

What is the inverse of the matrix below?

` [[8, 2], [6, 4]]`

Solution:

Inverse of the matrix is the

Inverse A = `1/((8*4)-(6*2)) [[4,-2],[-6,8]]`

`= 1/(32-12) [[4,-2],[-6,8]]`

`= 1/20[[4,-2],[-6,8]]`

` = 1/20* [[4,-2],[-6,8]]`

`= [[4/20, -2/20],[-6/20,8/20]]`

`[[4/20, -2/20],[-6/20,8/20]]= [[1/5, -1/10],[-3/10,2/5]]`

Complex matrix inverse - Example 2:

Find the inverse Complex matrix of the matrix below?

`[[9, 3], [7, 5]]`

Solution:

Inverse of the matrix is the

Inverse `A = 1/(9*5)-(7*3) [[5,-3],[-7,9]]`

`= 1/(45-21) [[5,-3],[-7,9]]`

`= 1/24[[5,-3],[-7,9]]`

`= 1/24* [[5,-3],[-7,9]]`

`= [[5/24, -3/24],[-7/24,9/24]]`

`[[5/24, -3/24],[-7/24,9/24]] = [[5/24, -1/8],[-7/24,9/24]]`

Complex matrix inverse - Example 3:

What is the inverse of the matrix below?

`[[10, 4], [8, 6]]`

Solution:

Inverse of the matrix is the

Inverse `A = 1/(10*6)-(8*4) [[6,-4],[-8,10]]`

`= 1/(60-32) [[6,-4],[-8,10]]`

`= 1/28 [[6,-4],[-8,10]]`

`= 1/28* [[6,-4],[-8,10]]`

`= [[6/28, -4/28],[-8/28,10/28]]`

`[[6/28, -4/28],[-8/28,10/28]] = [[3/14, -1/7],[-2/7,5/14]]`