Matrix Mechanics

Question 1

Given matrices AB=C, what row and column of A are being muiltiplied to find C_m,n?

Answer 1

Row m of A and

Column n of B

Question 2

Given matrices GH = F,

what operation would be performed to find F_m.n?

Answer 2

You would take the dot product of Row m of G and Column n of H.

Question 3

Each value in a matrix is referred to as an ______

Answer 3

Entry

Question 4

Given matrices AB = C,

what is the entry by entry formula that would produce entry C_4,5?

Answer 4

C_{4,5 =} A_4,1C_1,5 + A_4,2C_2,5 + A_4,3 C_{3,5 +} ….

= Σ A_4,kC_k,5 for k from 1 to n ( the number or columns in A or rows in B

Question 5

If you multilply a matrix by a one column vector, what shape answer to you get?

Answer 5

A one column vector

Question 6

If you multilply a matrix by a two column vector, what shape answer to you get?

Answer 6

A two column vector

Question 7

What are the 5 methods to muliply matrices

Answer 7

1. Standard Entry by Entry
2. Column scaling Column = Column
3. Row scaling Row = Row
4. Blocking
5. Row by Column = list of matricies to sum

Question 8

What is the advantage of multiplying matrices using the Row by Column = Series of matrices to sum.

Answer 8

Each of the individual resulting matrices is made up of Linearly Dependent vectors.

Question 9

How do you do Column by Row matrix multiplication?

Answer 9

Multiply each Row of the left matrix by the corresponding Column of the right matrix. Each of these multiplications will produce a matrix made up of LD vectors which is the size of the final answer.

Add all of these matrices together to get the final answer.

Question 10

In the Scaling Rows by Rows = Rows method, which matrix contains the rows that will be the scalars and which contains the row vectors that will be scaled?

Answer 10

The scaling scalars come from the left matrix. <–

The rows to be scaled come from the right matrix. –>

Question 12

How do you find the Inverse of a matrix?

Answer 12

1. Augment the original matirix A with I
2. then reduce this augmented matrix to the identiy matrix while doing the same steps to the augmented I
3. the right hand I will become the inverse of A

Question 13

What is the fundamental formula relating a matrix, its inverse and the Identity matrix I?

Answer 13

A A^-1 = i = A^-1 A

In other words A inverse can be multiplied on either the left or the right of A to give I

Question 14

What is the inverse of the product of two matrices AB?

Answer 14

The inverse of the product of AB is B^-1A^-1

This is because matirx multiplication is associative - it doesn’t matter what order you do it in. So multiply the inner two terms (B B^-1) to get I (which can then disappear) and then do the same with the outer terms (A A^-1) which yields I

Question 15

What the transpose of the Identity Matrix I?

Answer 15

I^T = I

Question 16

What is (A^T)^-1 in terms of A^-1 ?

Answer 16

(A^T)^-1 = (A^-1)^T

Question 17

How do you create the inverse of a Pivot Elementary Matrix?

Answer 17

Just change the sign of the scaling factor in the Pivot Elementary Matrix