Overview
• Section 7.2 in the textbook:
– Adding & ONE ntial.html">Subtracting Matrices
– Scalar Multiplication of Matrices
– Multiplying Matrices
– Matrix Multiplication & Systems of Equations |
| Adding & Subtracting Matrices |
Adding & Subtracting Matrices
• Given two matrices A and B, the two
matrices must be of the SAME size m x n
in order to add or subtract them
– Both matrices MUST be of the same
dimension; otherwise, they CANNOT be
added or subtracted |
• To add or subtract two matrices A and B:
– Add or subtract corresponding elements of A and B
 |
Adding & Subtracting Matrices
(Example)
Ex 1: Given matrices A and B below, perform the
fol lowing ope rations : A + B, B + A, A – B, B – A
if possible
 |
Adding & Subtracting Matrices
(Example)
Ex 2: Given matrices A and B below, perform the
following operations : A + B, B + A, A – B, B – A
if possible
 |
| Scalar Multiplication |
Scalar Multiplication
• To multiply a matrix A by a scalar (constant) c,
multiply every element of A by c
 |
Scalar Multiplication (Example)
Ex 3: Given matrices A and B below, perform the
following operations: 2A – 3B, 5A + ½B
 |
| Multiplying Matrices |
Multiplying a Row Matrix and a
Column Matrix
• More often we want to multiply by matrices
instead of scalars
• Row Matrix: a matrix that contains only
ONE row
– Has dimensions of 1 x n
• Column Matrix: a matrix that contains
only ONE column
– Has dimensions of n x 1 |
• The product of a row matrix and a column matrix
has dimensions of 1 x 1
(1 x n)(n x 1)
Given  and
Then
 |
Multiplying Matrices
• Before multiplying two matrices A and B, their
dimensions MUST satisfy the following:
(m x p)(p x n)
• The number of columns in the first matrix
MUST be the SAME as the number of rows in
the second matrix
• Otherwise, the matrices CANNOT be multiplied
• The resulting matrix will have dimensions of
m x n
• Each entry in the resulting matrix consists of
multiplying a 1 x p row matrix and a p x 1
column matrix for a 1 x 1 result
– Takes practice – study the diagram on the next slide |
 |
Multiplying Matrices (Example)
Ex 4: Given the matrices below, find AB and BA
if possible
 |
Multiplying Matrices (Example)
Ex 5: Given the matrices below, find AB and BA
if possible
 |
Matrix Multiplication &
Systems of Equations |
Matrix Multiplication & Systems of
Equations
• A system of equations can be decomposed into
a coefficient matrix , a variable matrix , and a
constant matrix
– This decomposition will become
important when we talk about
the Inverse of a Matrix
– The variable and constant
matrices are column matrices
– How can we check that the
decomposition and the original
system of equations are
equivalent ?
 |
Matrix Multiplication & Systems of
Equations (Example)
Ex 6: Decompose the system of equations into a
coefficient matrix, a variable matrix, and a
constant matrix – do not solve :
 |
Matrix Multiplication & Systems of
Equations (Example)
Ex 7: Convert the following decomposition into a
system of equations:
 |
Summary
• After studying these slides, you should know
how to do the following:
– Add and subtract matrices when possible
– Multiply a matrix by a scalar
– Multiply two matrices when possible
– Decompose a system of equations into a coefficient
matrix, a variable matrix, and a constant matrix
– Convert a decomposition into a system of equations
• Additional Practice
– See the list of suggested problems for 7.2
• Next lesson
– Inverse of a Matrix (Section 7.3) |
