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

 ┌     ┐      │-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.