Inverse Matrix - her.itesm.mx · Inverse of a Matrix •A every square matrix mwhose determinant is...
Transcript of Inverse Matrix - her.itesm.mx · Inverse of a Matrix •A every square matrix mwhose determinant is...
Content
• Inverse of a matrix• Elementary Row Operations (or Column)• Inverse of a Matrix by Elementary Operations• Recursion Formula for Inverse of Matrices
Inverse of a Matrix
• A every square matrix whose determinant is non‐zero there is an inverse matrix, denoted by , which satisfies the relations
where is the identity matrix (diagonal matrix with 1).
Inverse of a Matrix
• Singular Matrix. Is one whose determinant is zero.• Non‐Singular Matrix. Is one whose determinant is nonzero.• A singular matrix has no inverse.• The inverse matrix can be represented as follows:
1
where:is the determinant od the matrix A
is the adjoint of the matrix A
Inverse of a Matrix
• The Adjoint is the transpose of the matrix of cofactors of .• Example, using a 3 x 3 matrix:
⇒
∴
where the cofactors are: 1; ; ;; ; ;; ;
Inverse of a Matrix
• Solution:• First determine if the matrix is non‐singular (a nonzero determinant of ).
2 1 7 5 1 4 0 7 3 1 3 0 5 3 13 0 ⇒ Non‐singular
• Find the cofactors of .1 7 5 1 12; 0 7 3 1 3; 0 5 3 1 3;4 7 5 3 13; 2 7 3 3 5; 2 5 3 4 2;
4 1 1 3 7; 2 1 0 3 2; 2 1 0 4 2
⇒12 3 313 5 27 2 2
Inverse of a Matrix
• Find the adjoint (transpose of the cofactors of ).
⇒12 13 73 5 23 2 2
• Then the inverse of matrix is:
⇒1 1
3
12 13 73 5 23 2 2
4133
73
153
23
123
23
Elementary Row Operations (or Column)
• Elementary matrices:a) identity matrix with row multiplied by the scalar b) identity matrix with rows and exchangedc) identity matrix with row replaced by the sum of row and
times row
Elementary Row Operations (or Column)
• Example using elementary matrices 3 x 3:
31 0 00 3 00 0 1
;1 0 00 0 10 1 0
; 51 5 00 1 00 0 1
Elementary Row Operations (or Column)
• Multiplying elementary matrices to matrix we obtain:a) matrix with row multiplied by the scalar b) matrix with rows and exchangedc) matrix with row replaced by the sum of row and
times row
Elementary Row Operations (or Column)
• Example using elementary matrices 3 x 3:
1 3 40 2 52 3 1
31 3 40 6 152 3 1
;1 3 42 3 10 2 5
;
51 13 290 2 52 3 1
Inverse of a Matrix by Elementary Operations
• If is an invertible matrix of , form the matrix of , | . After that, perform elementary row operations until the first
columns form a reduced matrix equal to . The last columns will
• Then:→ ⋯ → |
• If a matrix is not reduced to , then does not have inverse.
Inverse of a Matrix by Elementary Operations
• Example:Find the inverse of the matrix using elementary row operations.
1 0 24 2 11 2 10
Recursion Formula for Inverse of Matrices
• Step 1. Normalize the element multiplying the row of the augmented matrix by the reciprocal of element . If the element is zero, then its reciprocal is not defined. In this case, the row must be exchanged for any row which does not have a zero element . In practice, the line is replaced by the row, where , is the element of maximum magnitude in column , above or below the main column.
1, … ,
• Step 2.Make zeros elements of the column of the augmented matrix, replacing the row for the best combination of row and row .
; 1, … ,
Recursion Formula for Inverse of Matrices
• Repeat the above procedure in the augmented matrix such that the left side is an identity matrix. The matrix on the right side is the inverse matrix.
Computer Program 1
• Submit a computer program to perform matrix operations and the inverse of matrices:• Adding and Subtracting Matrices• Scalar product• Matrices product• Determinant of a Matrix• Matrix inversion
• The maximum dimension of the matrix should be 100• Hand over:
• Flowchart (printed)• Computational algorithm (printed)• Source Code (printed and file)• Executable (file)