
Skills
to Master
·
Multiply
matrices of various dimensions by a constant (scalar)
·
Add and subtract
matrices of various dimensions
·
Multiply 2x2
matrices
·
Use the
properties of matrix algebra
Matrix
Algebra
In
mathematics, a matrix is simply an array of numbers in rows and columns. Matrices can be used in various ways, but we
are interested in how matrices relate to systems of equations. However, for this brief introduction we will
only be concerned with learning what a matrix is, some properties of matrices,
and the math operations that can performed on matrices.
A
matrix is a rectangular array of mn numbers arranged in m rows and n columns. We say that a
matrix is of size
.
Matrices
are usually represented by capital letters, but the following notations are
also common:
or 
In
the second representation, the matrix is an
matrix and the dotted
lines represent omitted values. In an
actual matrix with numbers, all numbers would be shown. The first representation is more compact and
the values of i and j are all numbers for i from 1 to m and j from 1 to n.
Two
matrices A and B are equal if both are of the same size and if the values of the
corresponding elements are the same. For
example,
is equal to
, but
is not equal to
since the values in
position
are not the same.
Addition/Subtraction
of Matrices
Adding
or subtracting matrices is a simple matter of adding/subtracting the
corresponding elements. But only
matrices of the same dimensions may be added or subtracted.
Example
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Scalar
Multiplication
A
scalar is a real number. Multiplying a matrix by a scalar is similar
to distributing.
Example
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Multiplying
Matrices
Matrices
may only be multiplied if the number of columns of the first matrix is equal to
the number of rows of the second. It is
possible to multiply, for example, a 4x3 matrix and a 3x2 matrix in this
order. The result is a matrix with the
same rows as the first matrix and the same columns as the second. For the above example, the resulting matrix
is of size 4x2. But note that these same
two matrices cannot be multiplied in the reverse order. That is, it is not possible to multiply a 3x2
matrix by a 4x3 matrix since the number of columns of the first is not equal to
the number of rows of the second. What
this means is that matrix multiplication
is not commutative. Changing the
order of a matrix multiplication may either result in a different answer or in
no answer at all.
To
multiply two matrices:
1. Determine if the matrix multiplication is possible by looking at
the number of columns of the first matrix and the number of rows of the second
matrix.
2. If multiplication is possible, determine the size of the
resulting matrix. For example, if A is a
4x5 matrix and B is a 5x3 matrix, then the resulting matrix C will be a 4x3
matrix.
3. Set up the resulting matrix with empty spaces.
4. Each entry
in the resulting matrix C
is the sum of the products of the corresponding entries in the ith row of A and the jth column of B.
Example


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Note
that a matrix of size n x n is called a square matrix. It is
always possible to multiply two square matrices of the same size, but reversing
the order of the multiplication may result in a different answer. Also notice that in the last example, the
second matrix (with ones in the diagonal) results in a matrix identical to the
multiplier. This matrix is called the identity matrix. It works like 1 when multiplying. Also, in matrix addition, a matrix with zeros
in all rows and columns is called the zero
matrix. Adding a zero matrix to any
other matrix of the same
dimensions results in
the identical matrix. Other matrix
properties are as follows:
For matrices A, B, and
C, and for scalars a and b:
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