Matrices

Question 1

Dimensions of a matrix

Answer 1

Written as x X y
No. of rows x no. of columns

Question 2

Matrix vs column vector vs row vector

Answer 2

A matrix has 2+ rows and columns
A column vector has one column
A row vector has one row

Question 3

If the value of a variable is a matrix

Answer 3

Use bold, capital letters or a Tilda under the letter

Question 4

Adding/subtracting matrices

Answer 4

Must be the same dimensions meaning its conformable for addition
Add/subtract the elements in corresponding positions

Question 5

Scalar multiplication

Answer 5

Multiply everything in the matrix by the scalar

Question 6

Square matrix

Answer 6

Has the same amount of rows and columns

Question 7

Zero matrix

Answer 7

A matrix where all the elements are zero

Question 8

Identity matrix

Answer 8

A square matrix where the diagonal from the top left is filled with 1s, every other value is 0

Question 9

How to find an element in a multiplied matrix?

Answer 9

• Take the corresponding row from the first matrix
• Take the corresponding column from the second matrix
• Multiply first in row by first in column and so on
• Sum the answers

Question 10

What must be true to be able to multiply matrices

Answer 10

Columns in matrix 1 = rows in matrix 2 so its conformable under multiplication

Question 11

Does AB = BA for matrix multiplication?

Answer 11

No, as matrix multiplication is non-commutative.

Question 12

Dimensions of a multiplied matrix

Answer 12

Rows in a x columns in b

Question 13

What is the relation between the determinant/magnitude and the area scale factor of a matrix transformation?

Answer 13

The determinant of a matrix represents the area scale factor of the transformation. The area of the image can
be found by multiplying the area of the object by the determinant.

Question 14

How to write determinant of matrix A

Answer 14

det(A) or |A|

Question 15

If det(A) = 0

Answer 15

A is a singular matrix and doesn’t have an inverse

Question 16

If det(A) is not equal to 0

Answer 16

A is non-singular and does have an inverse

Question 17

Determinant of matrix a b
c d

Answer 17

ad-bc

Question 18

Determinant of matrix a b c
d e f
g h i

Answer 18

a | e f | - b | d f | + c | d e |
| h i | | g i | | g h |

Do the 2x2 determinant of matrices not in the same row/column as the factor

Question 19

The minor of an element in a 3x3 matrix

Answer 19

The determinant of the matrix remaining when the row/column it is in are eliminated

Question 20

Inverse of a matrix

Answer 20

Written as M^-1, undoes the effect of a matrix
MM^-1 = M^-1M = I

Question 21

Inverse of a 2x2 matrix

Answer 21

1 ( d -b)
———-
det(A) ( -c a)

Question 22

What must you do when multiplying by an inverse matrix

Answer 22

Do it to the front of both sides of the equation or the back of both
Write that MM^-1 = I before removing

Question 23

A^T

Answer 23

Transpose of matrix A, inverted rows and columns, keeping the leading diagonal entires the same.

Question 24

Inverting a general matrix

Answer 24

1) find det(A)
2) make a matrix of minors
3) use that matrix with each second sign reversed
4) transpose it
5) multiply by 1/det(A)

Question 25

What are the three configuration of planes, if there are no solutions to the simultaneous equations?

Answer 25

Question 26

What is the one configuration of planes, if there are infinitely many solutions to the simultaneous equations?

Answer 26

Question 27

What is the one configuration of planes, if there is one unique solution?

Answer 27

Question 28

How can you spot when planes are parallel?

Answer 28

If two or more planes are parallel, they can be written so that the coefficients of x, y and z are the same but the constant term is different in Cartesian form.

e.g.:
x + 2y + 3z = 1 and x + 2y + 3z = 2 are parallel planes

x + 2y + 3z = 1 and 2x + 4y + 6z = 2 are the same plane

Question 29

What is true about the configuration of the planes, if the determinant is zero and none of the planes are parallel?

Answer 29

This means there is no unique solution, and if none of the planes are parallel, there are two possibilities:

• The equations are consistent, in which case there are infinitely many solutions and the planes for a sheaf.
• The equations are inconsistent, in which case there are no solutions and the planes must form a triangular prism.

Question 30

How do you distinguish the two possibilities if the determinant is zero for simultaneous equations?

Answer 30

Eliminate one of the variables so that you have two equations in two unknowns. Then check if the two equations are consistent or not (represent the same equation).

Question 31

How do you find the equation of the line that represents a sheaf in simultaneous equations?

Answer 31

Set one variable as a parameter in the three equations, then solve them simultaneously to find a vector equation of a line for x y and z.

e.g.

Question 32

What does conformability mean in matrices?

Answer 32

It means the dimensions are suitable for defining some operation (e.g. addition or multiplication)

Question 33

What are the steps to find invariant points or lines of invariant points of a matrix, T

Answer 33

T (x) = (x)
(y) = (y)
then solve for x and y

Question 34

What are the steps to find invariant lines of a matrix, T?

Answer 34

T (x) = (x’)
(y) = (y’)

then substitute y = mx+c after the matrix multiplication.

then produce an equation: y’ = mx’ + c by subbing in equations found

then solve for coefficients of x and then the term that includes c.