# Matrix Operations

Recall how the following matrix operations are defined.

• Scalar Multiplication

When and
then is defined by

for each 1 ≤ i ≤ m, 1 ≤ j ≤ n.

• Matrix Addition

When both and
then C = A + B is defined by

for each 1 ≤ i ≤ m, 1 ≤ j ≤ n.

Note that addition is only defined when both matrices have
the same number of rows and the same number

of columns.

• Matrix Multiplication

When and
then C = AB is defined by

for each 1 ≤ i ≤ m, 1 ≤ j ≤ n.

Note that matrix multiplication is only defined when the
number of columns of the first matrix is equal to

the number of rows of the second. Also observe that

• Matrix Transpose

When then
is defined by

or each 1 ≤ i ≤ m, 1 ≤ j ≤ n.

Note that

1. Consider the following matrices.

Calculate the following.

(a) A + B (b) B + C (c) − A (d) 2(A + C)

2. Consider the following matrices.

Calculate the following.

(a) AB (b) BA (c) AD (d) BC

(e) EF (f) FE (g) GH (h) HG

(i) EA (j) AH (k) EC (l) GD

3. For the matrices defined in exercise 2, compute the
following.

4. Given that the matrix product AB is defined, for scalar prove that

5. Given that the matrix sum A + B is defined, for scalar prove that

6. Given that the matrix sum A+B is defined and the matrix product CA is
defined, prove that C(A+B) =

CA + CB.

7. Given that the matrix product AB is defined, prove that

8. Let and consider the following three
functions defined on R^3 .

(a) F : R^3 --> what? G : R^3 --> what? H : R^3 --> what?

(b) Show all three of these functions are linear.

(c) It is possible to write each of these functions in the form of a matrix
product Mx. Explicitly determine

these three matrices.

9. Write the given systems of equations in matrix form,
i.e. y = Mx, for appropriate matrix M.

10. Consider a given matrix Show the
following sets of vectors both define a vector space.

Later we will call N the null space of the matrix M, and R will be called M’s range space.