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

M1M1

M1M2

M1M3

M2M3

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

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

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

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

6.       Find two square matrices, A and B, where neither A or B contains any 0s, yet AB 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 (AB = I) then (BA = I)
d) (AB)C = A(BC) [ 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.