8.2 OPERATIONS WITH MATRICES
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8.2 OPERATIONS WITH MATRICES Copyright © Cengage Learning. All rights reserved.
Transcript of 8.2 OPERATIONS WITH MATRICES
8.2 OPERATIONS WITH MATRICES
Copyright © Cengage Learning. All rights reserved.
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• 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.
What You Should Learn
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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
d. The sum of
and
is undefined because A is of order 3 3 and B is of
order 3 2.
cont’d
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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 m n Matrices
(Solve for x.) (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 and
Solution:
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For most matrices, AB BA.
Matrix Multiplication
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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 In A = A.
and
AI = A
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
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
cont’d
