Matrix Quick Study Guide

Vocabulary:

Augmented matrix	Columns	Commensurate matrices
Commutative Property	Dimensions	Elementary Row Operations
Elements	Identity matrix	Incommensurate matrices
Inverse	Invertible matrix	Main Diagonal
Matrix Addition	Matrix Multiplication	Matrix Subtraction
Non-invertible matrix	Non-singular matrix	Pivot
Product Matrix	Rectangular matrix	Rows
Scalar Multiplication	Singular matrix	Square matrix
Zero Matrix

Tasks:

┌ ┐ │-1 2│ M1 = │ 2 -5│ └ ┘

┌ ┐ │5 6 7│ M2 = │8 9 0│ └ ┘

┌ ┐ │1 2│ M3 = │3 4│ │5 6│ └ ┘

┌ ┐ │-5 -2│ M4 = │-2 -1│ └ ┘

1. Find each of the following (or mark them as incommensurate):

M1+M1

M1+M2

M1+M3

M1+M4

M1+2M4

M1×M1

M1×M2

M1×M3

M2×M3

2. Write I₄ (the 4x4 identity matrix).

3. Write a singular matrix that does not contain any 0’s.

4. Find two square matrices A and B that together show that matrix multiplication is non-commutative by showing that A×B ≠ B×A.

5. Find two square matrices, A and B, A≠B, where neither A or B is the identity matrix, yet A×B = B×A.

6. Find two square matrices, A and B, where neither A or B contains any 0’s, yet A×B is the zero matrix (all zeroes).

7. Without actually inverting either matrix, show that M₁ and M₄ are inverses of each other.

8. Prove each of the following statements (your proof may assume the 2x2 case):
a) I = I^-1.
b) if (A = B^-1) then (B = A^-1)
c) if (A×B = I) then (B×A = I)
d) (A×B)×C = A×(B×C) [ this is tedious ]

9. Manually invert the matrix M₁ (label each step as an elementary row operation).

10. Write a non-singular 3x3 matrix and manually invert it.

11. Prove that, in general, each Elementary Row Operation is really a Matrix Multiplication.

12. In class, we learned how to invert a matrix by constructing an augmented matrix and transforming the left-hand-side into the identity matrix using elementary row operations, at which point the right-hand-side is the inverse. Prove that this works in general.

13. Solve the following system of linear equations using matrix inversion on your calculator:

3x + 2y – 5z = 22
8x – 5y = -3
x + 3y – 7z = 8

1. Using matrix inversion on your calculator, find the equation of a polynomial that contains the points (-1,2), (0,1), (1,4), and (2,17). [ Hint: In general, you can fit k points with a (k-1)-degree polynomial. Here, we have 4 points, so we use a polynomial of degree 3. That is, a cubic polynomial, of the form: ax³ + bx² + cx + d.]

2. Using matrix inversion on your calculator, prove: 1⁵ + … + n⁵ = (1/12)(2n⁶ + 6n⁵ + 5n⁴ – n²). Then, use your calculator to find 1⁵ + … + 7⁵ and show that this formula works when n=7.