Download Introduction to Matrices

Introduction to
Matrices
HONORS ADVANCED ALGEBRA II/TRIGONOMETRY
MS. LEE
Essential Stuff

Essential Question: What is a matrix, and how do
we perform mathematical operations on
matrices?

Essential Vocabulary:
 Matrix
 Scalar
 Determinant
 Inverse
What is a matrix?

A matrix is a rectangular array of numbers, symbols, or
expressions arranged in rows and columns enclosed in a single
set of brackets.
A=
𝑎1,1
𝑎2,1
𝑎1,2
𝑎2,2

The dimensions of a matrix are the number of horizontal
rows and the number of vertical columns it has.

NOTE!!: The number of rows always comes before the
number of columns.
Matrix Terminology

Each number, expression, or symbol in a matrix is called
an element or an entry.
B=

4 −7
11 0
3
6
4ЄB
Entries are denoted by a variable and two subscripts
(rows and columns).

b1,2 = -7
b3,2 = 6
b2,1=
4=
Adding and Subtracting Matrices

You can add or subtract matrices if and only if
they have the same dimensions.

In order to add or subtract two or more matrices,
add their corresponding elements.
Adding and Subtracting Matrices
A
+
B
=
A+B
𝑒
𝑏
+
𝑔
𝑑
𝑎
𝑐
A
𝑎
𝑐
-
𝑒
𝑏
−
𝑔
𝑑
𝑓
𝑎+𝑒
=
ℎ
𝑐+𝑔
B
=
𝑏+𝑓
𝑑+ℎ
A-B
𝑓
𝑎−𝑒
=
ℎ
𝑐−𝑔
𝑏−𝑓
𝑑−ℎ
Examples
Scalar Multiplication

You can multiply a matrix by a constant called a
scalar.

In order to perform scalar multiplication on a
matrix, multiply each element in the matrix by
the scalar.
Examples
Homework
Homework
2.1
Matrix Multiplication

Matrix multiplication has no operational counterpart in
the real number system.

In order to multiply two matrices (matrix A and matrix B),
the number of columns in A must be equal to the
number of rows in matrix B.
Matrix A
Matrix B
Matrix AB
3x2
2x4
3x4
equal
Dimensions of AB
Matrix Multiplication

When multiplying two matrices, A and B, multiply
the entries of the first row of matrix A and the first
column of matrix B, then add those products up
to make the first entry in matrix AB.

Repeat this step until we have multiplied each
row in matrix A with each column in matrix B.
Examples
Determinant

The determinant is a real number associated with
SQUARE matrices. It tells us special things about the
matrix useful in solving systems of equations, calculus,
and more.

Notation: det(A) = |A|
𝑎
𝑐
𝑏
𝑑
, then det(A) = |A|=
𝒂
𝒄
𝒃
𝒅

Let A =
= ad – bc.

For any matrix larger than 2x2, the determinant will be
found using calculator.
Examples
Inverse of Matrices:

Not every matrix has an inverse.

A matrix has an inverse if and only if it's
determinant is not 0.

A matrix is invertible if and only if ad – bc ≠ 0.
𝑎
Let A =
𝑐
𝑏
be invertible.
𝑑
Then,
1
𝑎𝑑−𝑏𝑐
𝐴−1
=
𝑑
−𝑐
−𝑏
and 𝐴𝐴−1 = 𝐴−1 𝐴 = 𝐼
𝑎
Homework
 Homework
2.2