2
SYSTEMS OF
LINEAR
EQUATIONS AND
MATRICES
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2.4
Matrices
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Addition and Subtraction
3
Addition and Subtraction
Two matrices A and B of the same size can be added or
subtracted to produce a matrix of the same size. This is
done by adding or subtracting the corresponding entries in
the two matrices.
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Addition and Subtraction
The following laws hold for matrix addition.
The commutative law for matrix addition states that the
order in which matrix addition is performed is immaterial.
The associative law states that, when adding three
matrices together, we may first add A and B and then add
the resulting sum to C. Equivalently, we can add A to the
sum of B and C.
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Example 4
Let
a. Show that A + B = B + A.
b. Show that (A + B) + C = A + (B + C).
Solution:
a.
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Example 4 – Solution
cont’d
On the other hand,
7
Example 4 – Solution
cont’d
so A + B = B + A, as was to be shown.
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Example 4 – Solution
cont’d
b. Using the results of part (a), we have
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Example 4 – Solution
cont’d
Next,
so
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Example 4 – Solution
cont’d
This shows that (A + B) + C = A + (B + C).
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Addition and Subtraction
A zero matrix is one in which all entries are zero. A zero
matrix O has the property that
A+O=O+A=A
for any matrix A having the same size as that of O. For
example, the zero matrix of size 3  2 is
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Addition and Subtraction
If A is any 3  2 matrix, then
where aij denotes the entry in the ith row and jth column of
the matrix A.
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Addition and Subtraction
The matrix that is obtained by interchanging the rows and
columns of a given matrix A is called the transpose of A
and is denoted AT. For example, if
then
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Addition and Subtraction
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Scalar Multiplication
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Scalar Multiplication
A matrix A may be multiplied by a real number, called a
scalar in the context of matrix algebra. The scalar product,
denoted by cA, is a matrix obtained by multiplying each
entry of A by c.
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Example 5
Given
find the matrix X satisfying the matrix equation 2X + B = 3A.
Solution:
From the given equation 2X + B = 3A, we find that
2X = 3A – B
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Example 5 – Solution
cont’d
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Practice
p. 109 Exercises #4-6 transpose only, 8-12
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