Introduction to matrix inverse rules
The matrix (in singular) or matrices (in plural) is the rectangular array of numbers, letters. An item in a matrix is an element or an array. The identity matrix is a square matrix with all the entries as zero. Elements in the matrix are denoted by the letter with two subscripts as A11. Additive identity matrix denotes the identity matrix with m rows and m columns. The Identity matrix is denoted by letter O. The 1Ół in leading diagonal and all others are zero in a matrix is the identity matrix.
Properties of Matrix Inverse Rules
X is invertible.
X is row-equivalent to the a-by-a identity matrix In.
X is column-equivalent to the a-by-a identity matrix In.
Rank of a matrix X = a.
The vertical entries(columns) of matrix X are linearly independent.
The columns of X span Kn (i.e., Column X = Kn).
The columns of X form a basis of Kn.
There is an a by a matrix Y such that XY = In.
There is an a by a matrix Z such that Z X = In.
The transpose XT is an invertible matrix (hence rows of X are linearly independent, span Kn, and form a basis of Kn).
The number 0 is not an eigenvalue of X.
The matrix X can be denoted as a finite product of elementary matrices.
Algebra is widely used in day to day activities watch out for my forthcoming posts on how to simplify algebraic fractions and algebra 2 equation solver. I am sure they will be helpful.
Example - Matrix Inverse Rules
Problem 1: Compute the inverse for the matrix.
`[[1,3,1],[1,1,2],[2,3,4]]`
Solution: Finding determinant of A
|A| = 1x(1x4-3x2) - 3x(1x4-2x2) + 1x(1x3-2x1)
|A| = 1x(4-6) - 3x(4-4) + 1x(3-2) = -2+0+1
|A| = -1
Finding Minors of A:
M11 = 1x4-3x2 = 4-6 = -2
M12 = 1x4-2x2 = 4-4 = 0
M13 = 1x3-2x1 = 3-2 = 1
M21 = 3x4-3x1 = 12-3 = 9
M22 = 1x4-2x1 = 4-2 = 2
M23 = 1x3-2x3 = 3-6 = -3
M31 = 3x2-1x1 = 6-1 = 5
M32 = 1x2-1x1 = 2-1 = 1
M33 = 1x1-1x3 = 1-3 = -2
Forming Minors Matrix of A:
` [[-2,0,1],[9,2,-3],[5,1,-2]]`
Forming Cofactor Matrix of A:
`[[-2*1,0*1,1*1],[9*1,2*1,-3*1],[5*1,1*1,-2*1]]` = `[[-2,0,1],[-9,2,3],[5,-1,-2]]`
Forming Adjoint A:
Adj(A) = `[[-2,-9,5],[0,2,-1],[1,3,-2]]`
Finding the Inverse Matrix of A
Inverse of a matrix = A-1 = `(Adj(a)) / (Det(A))` = `1 / -1` = `[[-2,-9,5],[0,2,-1],[1,3,-2]]`
A-1 = `[[2,9,-5],[0,-2,1],[-1,-3,2]]`
The matrix (in singular) or matrices (in plural) is the rectangular array of numbers, letters. An item in a matrix is an element or an array. The identity matrix is a square matrix with all the entries as zero. Elements in the matrix are denoted by the letter with two subscripts as A11. Additive identity matrix denotes the identity matrix with m rows and m columns. The Identity matrix is denoted by letter O. The 1Ół in leading diagonal and all others are zero in a matrix is the identity matrix.
Properties of Matrix Inverse Rules
X is invertible.
X is row-equivalent to the a-by-a identity matrix In.
X is column-equivalent to the a-by-a identity matrix In.
Rank of a matrix X = a.
The vertical entries(columns) of matrix X are linearly independent.
The columns of X span Kn (i.e., Column X = Kn).
The columns of X form a basis of Kn.
There is an a by a matrix Y such that XY = In.
There is an a by a matrix Z such that Z X = In.
The transpose XT is an invertible matrix (hence rows of X are linearly independent, span Kn, and form a basis of Kn).
The number 0 is not an eigenvalue of X.
The matrix X can be denoted as a finite product of elementary matrices.
Algebra is widely used in day to day activities watch out for my forthcoming posts on how to simplify algebraic fractions and algebra 2 equation solver. I am sure they will be helpful.
Example - Matrix Inverse Rules
Problem 1: Compute the inverse for the matrix.
`[[1,3,1],[1,1,2],[2,3,4]]`
Solution: Finding determinant of A
|A| = 1x(1x4-3x2) - 3x(1x4-2x2) + 1x(1x3-2x1)
|A| = 1x(4-6) - 3x(4-4) + 1x(3-2) = -2+0+1
|A| = -1
Finding Minors of A:
M11 = 1x4-3x2 = 4-6 = -2
M12 = 1x4-2x2 = 4-4 = 0
M13 = 1x3-2x1 = 3-2 = 1
M21 = 3x4-3x1 = 12-3 = 9
M22 = 1x4-2x1 = 4-2 = 2
M23 = 1x3-2x3 = 3-6 = -3
M31 = 3x2-1x1 = 6-1 = 5
M32 = 1x2-1x1 = 2-1 = 1
M33 = 1x1-1x3 = 1-3 = -2
Forming Minors Matrix of A:
` [[-2,0,1],[9,2,-3],[5,1,-2]]`
Forming Cofactor Matrix of A:
`[[-2*1,0*1,1*1],[9*1,2*1,-3*1],[5*1,1*1,-2*1]]` = `[[-2,0,1],[-9,2,3],[5,-1,-2]]`
Forming Adjoint A:
Adj(A) = `[[-2,-9,5],[0,2,-1],[1,3,-2]]`
Finding the Inverse Matrix of A
Inverse of a matrix = A-1 = `(Adj(a)) / (Det(A))` = `1 / -1` = `[[-2,-9,5],[0,2,-1],[1,3,-2]]`
A-1 = `[[2,9,-5],[0,-2,1],[-1,-3,2]]`
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