Matrix Algebra and Inverses

Transcription

1 restart; with(inearalgebra): We start with two random 3x3 matrices by using the Randomatrix command. A:=Randomatrix(3,3,generator=-9..9); A := B:=Randomatrix(3,3,generator=-9..9); B := Checking equality of matrices with the Equal command. Equal(A,B); false atrix addition (three ways). A+B; Add(A,B); atrixadd(a,b); pposites and scalar multiplication (two ways). -A; *A; atrix Algebra and nverses

2 Scalarultiply(A,-3); s -A = (-1)A? Equal(-A,-1*A); atrix subtraction. A-B; Add(A,-B); atrixadd(a,-b); A-*B; Add(A,-*B); atrixadd(a,-*b); true

3 atrix multiplication (two ways). A:=Randomatrix(4,2,generator=-9..9); 6 1 A := 8 4 B:=Randomatrix(2,3,generator=-9..9); B := ultiply(a,b); atrixatrixultiply(a,b); atrix multiplication is not commutative, atrixatrixultiply(b,a); Error, (in inearalgebra:-atrixatrixultiply) first matrix column dimension (3) <> second matrix row dimension (4) The multiplication cannot even be done due to incompatible matrix dimensions. We look at matrix multiplication symbolically. A:=atrix(3,3,symbol=a); a 1, 1 a 1, 2 a 1, 3 A := a 2, 1 a 2, 2 a 2, 3 a 3, 1 a 3, 2 a 3, 3 B:=atrix(3,3,symbol=b); b 1, 1 b 1, 2 b 1, 3 B := b 2, 1 b 2, 2 b 2, 3 b 3, 1 b 3, 2 b 3, 3 ultiply(a,b);

4 a 1, 1 b 1, 1 C a 1, 2 b 2, 1 C a 1, 3 b 3, 1 a 1, 1 b 1, 2 C a 1, 2 b 2, 2 C a 1, 3 b 3, 2 a 1, 1 b 1, 3 C a 1, 2 b 2, 3 C a 1, 3 b 3, 3 a 2, 1 b 1, 1 C a 2, 2 b 2, 1 C a 2, 3 b 3, 1 a 2, 1 b 1, 2 C a 2, 2 b 2, 2 C a 2, 3 b 3, 2 a 2, 1 b 1, 3 C a 2, 2 b 2, 3 C a 2, 3 b 3, 3 a 3, 1 b 1, 1 C a 3, 2 b 2, 1 C a 3, 3 b 3, 1 a 3, 1 b 1, 2 C a 3, 2 b 2, 2 C a 3, 3 b 3, 2 a 3, 1 b 1, 3 C a 3, 2 b 2, 3 C a 3, 3 b 3, 3 Finding inverses by the aple procedure atrixnverse. A:=Randomatrix(4,4,generator=-9..9); A := B:=atrixnverse(A); B := atrixatrixultiply(a,b); atrixatrixultiply(b,a); ow we find the inverse using Gauss-ordan elimination or ReducedRowEchelonForm. We augment columnwise the square matrix A with the identity matrix of the same size.

5 4:= atrix(4,4,shape=identity); := C:=<A 4>; C := To augment rowwise, replace the bar with a comma. We put the augmented matrix in reduced row echelon form. C1:=ReducedRowEchelonForm(C); C1 := The right half Subatrix is the inverse. B:=Subatrix(C1,[1..4],[..8]); B := The inverse can be found using Gaussian elimination, with possible partial pivoting. C1:=GaussianElimination(C);

6 C1 := We now build the inverse matrix one column at a time by using back substitution. The above matrix can also be viewed as the matrix for four systems of four equations in four unknowns, all having the same coefficient matrix, found in the first four columns. Each of columns thru 8 contains the right hand side for one of the four systems. We first find column 1 of the inverse, which is also the solution vector for the first system of equations. col1:=backwardsubstitute(subatrix(c1,[1..4],[1..])); col1 := We find column 2 of the inverse, which is also the solution vector for the second system of equations. col2:=backwardsubstitute(subatrix(c1,[1..4],[1..4,6])); col2 := We find column 3 of the inverse, which is also the solution vector for the third system of equations. col3:=backwardsubstitute(subatrix(c1,[1..4],[1..4,7]));

7 col3 := We find column 4 of the inverse, which is also the solution vector for the fourth system of equations. col4:=backwardsubstitute(subatrix(c1,[1..4],[1..4,8])); col4 := We now build the inverse matrix from its columns. B:=<col1 col2 col3 col4>; B := atrixatrixultiply(a,b); atrixatrixultiply(b,a);

8 We find the Transpose of B. E:=Transpose(B); We can find the inverse B by solving the matrix equation AB = identity. B:=inearSolve(A,4); B := E := Finding the determinant of a square matrix by using the built-in Determinant function. Determinant(B); 1 Determinant(E); 1 t seems that a matrix and its transpose have the same determinant. Using Gaussian elimination. G:=GaussianElimination(B);

9 Since the matrix is triangular, the determinant is the product of the diagonal elements. d:=1:for i from 1 to 4 do d:=d*g[i,i] end do:d; 1 G :=