1. Define the following matrices

m1 = 3×4 matrix of zeroes
m2 = the 3×3 matrix where row1=(1,2,0), row2=(2,5,-1), row3=(4,10,-1)
m3 = transpose of m2
m4 = matrix product of m2 and m3
m5 = sum of m2 and m3
m6 = elementwise product of m2 and m3
m7 = matrix inverse of m2
m8 = the 5×5 identity matrix


m1 = Table[0, {3},{4}]
m2 = { {1,2,0}, {2,5,-1}, {4,10,-1} }
m3 = Transpose[m2]
m4 = m2 . m3
m5 = m2 + m3
m6 = m2 * m3
m7 = Inverse[m2]
m8 = IdentityMatrix[5]


2: Eigenvalues, Eigenvectors, SVD

Perform these essential matrix functions on matrix M:
M   =

 5.2  3.0
-4.1 -3.9

e = eigenvalues of M
v = eigenvectors of M
SVD of M  
(singular value decomposition)

m = { {5.2,3.0}, {-4.1,-3.9} }
e = Eigenvalues[m]
v = Eigenvectors[m]
SingularValues[m]


3: Solve matrix equation Ax=b

Solve Ax=b (and check the solution) given that

A =
1   2   0
2   5  -1
4  10  -1
      b =
7
8
9


A = { {1,2,0}, {2,5,-1}, {4,10,-1} }
b = {7,8,9}
x = LinearSolve[ A, b ]
A . x (* yes, it does equal b *)