Download 8.2 operations with matrices

8.2
OPERATIONS WITH MATRICES
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What You Should Learn
• Decide whether two matrices are equal.
• Add and subtract matrices and multiply matrices
by scalars.
• Multiply two matrices.
• Use matrix operations to model and solve
real-life problems.
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Equality of Matrices
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Equality of Matrices
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Equality of Matrices
Two matrices A = [aij] and B = [bij] are equal if they have
the same order (m  n) and aij = bij for 1  i  m and
1  j  n.
In other words, two matrices are equal if their
corresponding entries are equal.
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Example 1 – Equality of Matrices
Solve for a11, a12, a21, and a22 in the following matrix
equation.
Solution:
a11 = 2, a12 = –1, a21 = –3, and a22 = 0.
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Matrix Addition and Scalar
Multiplication
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Matrix Addition and Scalar Multiplication
Matrix Subtraction
A – B = A + (-B)
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Example 2 – Addition of Matrices
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Example 2 – Addition of Matrices
cont’d
d. The sum of
and
is undefined because A is of order 3  3 and B is of
order 3  2.
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Matrix Addition and Scalar Multiplication
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Matrix Addition and Scalar Multiplication
If A is an m  n matrix and O is the m  n zero
matrix consisting entirely of zeros, then A + O = A.
O is the additive identity for the set of all m  n matrices.
2  3 zero matrix
2  2 zero matrix
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Matrix Addition and Scalar Multiplication
Real Numbers
(Solve for x.)
m  n Matrices
(Solve for X.)
x + a= b
X+A=B
x + a + (–a) = b + (–a)
X + A + (–A) = B + (–A)
x+0 =b–a
X+O=B–A
x=b–a
X=B–A
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Matrix Multiplication
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Matrix Multiplication
For the product of two matrices to be defined, the number of columns of
the first matrix must equal the number of rows of the second matrix.
A

B
=
AB
mn
np
mp
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Example 7 – Finding the Product of Two Matrices
Find the product AB using
Solution:
and
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Matrix Multiplication
For most matrices, AB  BA.
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Matrix Multiplication
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Matrix Multiplication
If A is an n  n matrix, the identity matrix has the property
that AIn = A and InA = A.
AI = A
and
IA = A
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Applications
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Applications
A

X
=
B
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Example 11 – Solving a System of Linear Equations
Consider the following system of linear equations.
x1 – 2x2 + x3 = – 4
x2 + 2x3 = 4
2x1 + 3x2 – x3 = 2
a. Write this system as a matrix equation, AX = B.
b. Use Gauss-Jordan elimination on the augmented matrix
to solve for the matrix X.
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Example 11 – Solution
a. In matrix form, AX = B, the system can be written as
follows.
b. The augmented matrix is formed by adjoining
matrix B to matrix A.
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Example 11 – Solution
cont’d
Using Gauss-Jordan elimination, you can rewrite this
equation as
So, the solution of the system of linear equations is
x1 = –1, x2 = 2, and x3 = 1, and the solution of the matrix
equation is
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