MATRICESMATRICES

Adapted from presentation found on the internet. Thank you to the creator of the original presentation!

What is a Matrix?What is a Matrix?

• A matrix is a set of elements, organized into rows and columns

dc

barows

columns

Matrix EssentialsMatrix Essentials

•Plural form is matrices.•The “things” in a matrix are called elements or members.

Matrix Essentials Matrix Essentials (continued)(continued)

• The dimensions of a matrix are determined by rows x columns.– C3x4 means that matrix C has 3 rows and

4 columns– E5x2 means that matrix E has 5 rows and

2 columns

• C3x4 is different from C3,4 which means the element in the 3rd row and 4th column of matrix C.

Element IdentificationElement Identification

E3x5 = [ ] Identify the requested

elements.

18 -2 4 6 0

3 5 -1 7 23

-9 11 8 16 1

Row MatrixRow Matrix

•Row matrix – matrix having only 1 row– EX: B1x3 = [2 4 6]

Column MatrixColumn Matrix

•Column matrix– matrix having only 1 column

–EX: D2x1 = [ ]3

8

Square MatrixSquare Matrix

• Matrix in which the number of rows is the same as the number of columns.

• There are three things that only exist with a square matrix (more on these later):– determinant– identity matrix– inverse matrix

Equivalent MatricesEquivalent Matrices

•Two things must occur to have equivalent matrices:– They have the same dimensions.

– Corresponding elements have the same value.

Addition and SubtractionAddition and Subtraction

• Addition and subtraction can only be done if the dimensions of the two matrices match.– A 3x4 matrix can be added to another

3x4 matrix, BUT …– A 2x6 matrix cannot be added to a 2x4

matrix. (# of columns does not match)– A 3x2 matrix cannot be added to a 2x3

matrix. (Neither dimension matches. Don’t get tripped up by this one.)

Addition and Subtraction Addition and Subtraction (cont.)(cont.)

hdgc

fbea

hg

fe

dc

ba

hdgc

fbea

hg

fe

dc

ba

Just add corresponding

elements

Just subtract corresponding

elements

Scalar MultiplicationScalar Multiplication

• Scalar multiplication involves nothing more than multiplying a number or variable outside a matrix by every term inside the matrix.

3[ ] =4 3 -27 5 9

[ ]12 9 -621 15 27

Matrix MultiplicationMatrix Multiplication

• Two matrices can be multiplied only if the number of columns from the 1st matrix equals the number of rows from the 2nd matrix.

• Can the following matrices be multiplied? – A2x3 and B3x4? – C3x1 and D3x1? – E3x2 and F2x3? – G2x2 and H2x2?

Size of Product MatrixSize of Product Matrix

• The product has the same number of rows as the first matrix and the same number of columns as the second matrix.

• Size is based on the outer dimensions.

Matrix Multiplication Matrix Multiplication with Square Matriceswith Square Matrices

dhcfdgce

bhafbgae

hg

fe

dc

ba

Multiply each element in a row by the corresponding element in the column of the second matrix.

Commutativity of Matrix Commutativity of Matrix MultiplicationMultiplication

• Is AB = BA? Maybe, but maybe not!• Multiplication may be possible but resulting

in different elements in the answer.

• Multiplication may be possible but resulting in different size product matrix– A3x1 ●B1x3 = P3x3 BUT . . .

– B1x3 ● A3x1 = P1x1

......

...bgae

hg

fe

dc

ba

......

...fcea

dc

ba

hg

fe

Commutativity (continued)Commutativity (continued)

• A third reason for matrix multiplication not to be commutative is that the multiplication may be possible in one direction but not in the other.– A2x4 ● B4x5 = P2x5 BUT …

– B4x5 ● A2x4 cannot be multiplied.

Determinant of a MatrixDeterminant of a Matrix

• Used for inversion• If det(A) = 0, then A has no inverse.• Square matrix always has determinant.

dc

baA

bcadA )det(

Symbol is

Determinant of a 3x3 Determinant of a 3x3 MatrixMatrix

cegbdiafhcdhbfgaei

ihg

fed

cba

ihg

fed

cba

ihg

fed

cba

ihg

fed

cbaSum from left to rightSubtract from right to leftNote: N! terms

Mr. Parker’s note: We will do ours in a less jumbled, more organized way.

Determinant of a 3x3 Determinant of a 3x3 MatrixMatrix

Rewrite first two columns. Find “down” total. Find “up” total.

Det = DOWN – UP

Determinants and Area of Determinants and Area of Triangles in the Coordinate Triangles in the Coordinate

PlanePlaneIf you know the coordinates of the vertices of a triangle in the coordinate plane, you can use the absolute value of a 3x3 determinant to find the area of the triangle.

Coordinates are (x1, y1)(x2, y2) and (x3, y3)

Area is | |12

x1 y1 1x2 y2 1x3 y3 1

Identity MatrixIdentity Matrix

100

010

001

I

• Identity matrix only exists with a square matrix.

• On principal diagonal the elements are 1s. Everywhere else is 0s.

• Identity matrix: AI = A

Inverse of a MatrixInverse of a Matrix

• When discussing inverse, it is implied that we are talking about the multiplicative inverse.

• Inverse only exists with square matrices.

• If the inverse exists, then: AA-1 = I• 2x2 inverse can be done easily by

hand. 3x3 inverse is better found by using technology.

Inverse of a 2x2 MatrixInverse of a 2x2 Matrix

ac

bd

bcadA

11

* As we said before, the inverse of A exists only as long as det A ≠ 0.

or

Inverse of a MatrixInverse of a Matrix

100

010

001

ihg

fed

cba

1. Append the identity matrix to A

2. Subtract multiples of the other rows from the first row to reduce the diagonal element to 1

3. Transform the identity matrix as you go

4. When the original matrix is the identity, the identity has become the inverse!