Transcript Matrices!!

Matrices!!
Chapter 4
Matrices

Matrix: A rectangular arrangement of data
into rows and columns, identified by
capital letters.
 Example: A
Dimensions

Matrix Dimensions: Number of rows, m,
by the number of columns, n. Read as “m
by n” matrix. Also known as the order of
a matrix.

RBC (ROWS BY COLUMNS)
Elements
Matrix Element: Each number in a matrix,
identified by its row and column.
Example: amn
Refers to the mth row and nth column
Other Definitions

Zero Matrix: The additive IDENTITY of
matrices. A matrix whose elements are
all zeros.

Equal Matrices: Matrices with the same
dimensions and equal corresponding
elements.

Scalar: A real number factor.
Adding and Subtracting Matrices
 When
matrices have the same
dimension you add and subtract
them by adding or subtracting each
corresponding element.
Examples
Add or Subtract.
Matrix Equation
Matrix equation is an equation in
which the variable is a matrix.
You can solve for the variable by
adding or subtracting a matrix a
matrix to both sides to an
equation.
MATRIX
MULTIPLICATION
Objective:
By the end of class student will be able
to multiply matrices.
Review of 4.1 and 4.2
Example
Determine the dimensions of each matrix.
Example
Identify each element.
1. a23
2. a12
3. a31
4. a21
Try Some!
1.
A+B
3.
4A
2.
C–B
4.
2B – C
Matrix Multiplication
Matrix Multiplication

When multiplying matrices A and B, the
number of COLUMNS in matrix A MUST
be equal to the ROWS in matrix B.
The size of the product is
# rows in A x
# columns in B.
Multiplying Matrices
Can the following Matrices be multiplied? If
so, what dimensions will the product be??
1. [ 2 1 -4 ] x é 3 -2 4 ù
ê
ú
ê -1 0 1 ú
ê 5 -5 2 ú
ë
û
Multiplying Matrices
Can the following Matrices be multiplied? If
so, what dimensions will the product be??
é 2 -1 -4 ù é 5
1. ë
ûx ê
ù
3 2
ú
ë 1 -1 -3 û
Multiplying Matrices, Try Some!
Can the following Matrices be multiplied? If
so, what dimensions will the product be??
1.
2.
3.
4.
How to multiply matrices
 Multiply
the elements of each row in
the first matrix by the elements in
each column of the second matrix
 Add
the products to get the new
element.
Example: Multiply the following.
Example: Multiply the following.
Example: Multiply the following.
Example: Multiply the following.
Example: Multiply the following.
Try Some!!
Try Some!!
DETERMINANT OF
MATRICES
Determinant of 2 x 2
Determinant of 2 x 2

Find the determinant of the following 2x2
matrices:
Determinant of 2 x 2

Find the determinant of the following 2x2
matrices:
Determinant of 3x3

To find the determinant of a 3x3 matrix
you still need to look at the the diagonals
to the left, and diagonals to the right.
 Step
1: Rewrite the first 2
columns
 Step 2: Multiply the diagonals
 Step 3: (sum of the left diagonals)
– (sum of the right diagonals)
Example
 Find
the determinant of the
following.
Try Some!
 Find
the determinant of the
following.
INVERSE OF
MATRICES
Inverse
REMEMBER we denote inverse
with a -1 power
So the inverse of matrix A is A-1
Requirement to have an Inverse
Matrix MUST be square,
meaning it has the same
number of rows and columns
Matrix MUST NOT have a
determinant of zero.
Inverse exist?!
Does the inverse exist?!?!
Multiplying Inverse
When you Multiply a matrix A times it’s
inverse. The Product is the Identity
Matrix.
Identity Matrix is a
square matrix where
the top left to
Bottom right
diagonal are all ones,
and everything else is
a zero
Inverse
Determine if the follow matrices are
inverses.
1. é
ùé
ù
11 -5
1 -5
ê
úê
ú
ë 2 -1 ûë 2 -11 û
2. é 0
-2 ùé 4 -1 ù
ê
úê
ú
ë -1 -9 ûë -2 0 û
Finding the Inverse of a 2x2
IF
THEN
Example
Find the inverse of the following matrix.
Use your calculator!
2nd  Matrix  Edit
Put in your matrix
2nd  Matrix  NAME
Get your matrix
X-1
Inverse Application
We can use matrix Inverse when solving
matrix equations.
For Matrices A and B, we can find Matrix X.
IF AX = B
THEN X = A-1B
Example
Solve for the Matrix X.
Try Some!
Solve each matrix equation.
Try Some!