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# 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.

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.