Start Maple from the desktop, Start Menu/Program Files/ or using the icon in f:\"math 1204". Note that if you load the Matrix Algebra programs using > with(LinearAlgebra): then the new functions aren't listed, but Maple still knows them, the colon just means to not print the result of the command. Define, > L:=Matrix([[4, -2, 0], [-1, 2, 2], [2, -2, 3]]); We can use Pivot or RowOperation to produce the inverse: Create > LI:=< L | IdentityMatrix(3) >; and use RowOperation to produce the Identity Matrix on the left where J was. You should get the same matrix as from > Li:=MatrixInverse(L); Note that the common denominator in the fractions in the inverse is 26, not 13, and -26 is the determinant: > Determinant(L); We can use > L12:=SubMatrix(L,[2,3],[1,3]); to find the cofactor submatrix when we remove row 1 and column 2. Note that the first square brackets list the rows we want to include and the second square brackets have the columns. Create L11 and L13 similarly and verify that > L[1,1]*Determinant(L11) - L[1,2]*Determinant(L12) + L[1,3]*Determinant(L13); gives the same as the determinant of L. Choose one of the three columns of L and do a cofactor expansion using it. Now we will try to find the inverse of a 4x4. > randomize(): K:=RandomMatrix(4,4,generator=rand(-3..3)); First see if K has determinant 0. > Determinant(K); If it does, repeat until you get a new K that will work. Use Pivot and row operations on > KI:=< K | IdentityMatrix(4) >; and compare your answer with > MatrixInverse(K); Now we will look at how Maple copes with multiple solutions. Create a random matrix J0 to represent a 3x4 matrix equation and its solution vector: > J0:=RandomMatrix(3,5,generator=rand(-4..4)); Check that > Rank(J0); is equal to 3, the same as the number of rows. Use Pivot and/or RowOperation to produce as many zeros and ones as possible in the first four columns. Because of the rank you will have just one column you couldn't pivot. Now use your brain and/or pen and paper to predict the particular and homogeneous solutions and input them using > h:= < number, number, number, number >; and > p:= < number, number, number, number >; with the "number"s being your found values. Note that you can extra the 3x4 matrix K as follows > J:=DeleteColumn(J0,5); and then do your checks with > Multiply(J,h); Multiply(J,p); Start again with the same J0 but leave a different column unpivoted this time. Continue until you again have a different particular solution and notice that the homogeneous solution involves the same ratios between numbers. Notice how similar answers appear with the commands: > ReducedRowEchelonForm(J0); and > LinearSolve(J0,free='t'); Create a random 4x3 matrix M using > M:=RandomMatrix(4,3,generator=rand(3..11)); and then create N as a random 3x4 matrix, similarly. Multiply M and N, firstly with > Multiply(M,N); and then with > Multiply(N,M); and notice that these two are not the same size nor have the same numbers in them; it is important to remember that it matters with matrices which order they are being multiplied in. Maple will refuse to multiply matrices of the wrong size, try to multiply M by itself, N by L and L by N and see if it works when you expect it to. The purple response tries to explain what went wrong.