Math1300:MainPage/Matrices
Contents
• 1 Matrices and Matrix Operations
♦ 1.1 Size
♦ 1.2 Equality
♦ 1.3 Addition and Subtraction of Matrices
♦ 1.4 Scalar Multiplication
♦ 1.5 Matrix Multiplication
♦ 1.6 Transpose, Trace and Diagonal
◊ 1.6.1 Theorem (Properties of the Transpose)
♦ 1.7 Linear Combinations of Columns
♦ 1.8 Linear Combinations, Systems of Linear Equations and Matrix
Multiplication
♦ 1.9 Properties of Matrix Arithmetic
◊ 1.9.1 Theorem (Properties of Addition and Scalar
Multiplication)
◊ 1.9.2 Theorem (Properties of Matrix Multiplication)
♦ 1.10 The zero matrix and the identity matrix In
◊ 1.10.1 Theorem (Properties of the Zero Matrix)
◊ 1.10.2 Theorem (Properties of the Identity Matrix)
♦ 1.11 Solving the Matrix Equation AX=B
Matrices and Matrix Operations
Size
An m by n matrix is a rectangular array of numbers with m rows and n columns. The numbers m and n define the
size of the matrix. When it is desirable to emphasize the size of the matrix, the following notation is used:
Such a matrix is called "m by n". This is written as
An
matrix is called square if m = n. To emphasize the size of a square matrix, a single subscript is used:
Matrices and Matrix Operations
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Equality
Two matrices A = [ai,j] and B = [bi,j] are equal if they are the same size, say m by n, and
a = b , for
i,j
and
i,j
Examples:
1.
2.
(note that this is the same as four equations in two unknowns;
they imply x = y = 1)
Addition and Subtraction of Matrices
For two matrices A = [ai,j] and B = [bi,j], addition is defined if and only if the matrices have the same size. In that
case, we say that the matrix C = [ci,j] satisfies C = A + B if and only if
ci,j = ai,j + bi,j
Similarly, for two matrices A and B of the same size, C = A − B is defined by
ci,j = ai,j − bi,j
Example: If
and
then
and
In short, addition and subtraction of two matrices are carried out by adding or subtracting the corresponding
positions within the matrices.
Scalar Multiplication
A scalar, in our context, is any real number. If A = [ai,j] is a matrix and r is a scalar, then the matrix C = rA is
defined by
ci,j = rai,j.
Equality
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Example: If
and r = 2, then
In short, the product rA is computed by multiplying every entry of A by r.
Matrix Multiplication
The definition of matrix multiplication is very different from that of addition and subtraction. Suppose we have
two matrices A and B with respective sizes
and
The product of A and B is defined only if n = r,
that is, the number of columns of A is equal to the number of rows of B. When this is the case, The matrix C =
[ci,j] is defined in the following way: consider the entries in row i of the matrix A:
and also
the entries in column j of the matrix B:
Then
Notice that the assumption n = r implies that there is just the right number of entries in the rows of A and columns
of B to allow ci,j to be defined. The number ci,j is also called the inner product of row i of A and column j of B.
Notice that this definition implies that the size of the product is
Examples:
1. Suppose
and
Then C = AB is defined and has size
.
. Here are the entries in C:
In other words
2. Using A and B from the previous example, the matrix D = BA is also defined. In this case the product is
of size
In this case we have
Note that
Scalar Multiplication
since the two matrices, in this case, have different size.
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3.
Let
and A as in the previous examples. Then
4.
Let
. Computing as in the last example, we have
AI3 = A.
Transpose, Trace and Diagonal
Suppose that A is a square matrix. Then the diagonal elements are the ones joining the upper left and lower right
entries of the matrix, that is, the entries of the form
The trace is the sum of the diagonal
entries.
The transpose of A is the matrix AT obtained by making the first row of A the first column of AT, the second row
of A the second column of AT, etc. Another way of saying this is that AT = [aj,i].
Examples:
1. Let
Then
The diagonal elements of A are 1, 5, and 9, while the trace of A is 1 + 5 + 9 = 15.
2. The identity matrix In of order n has all diagonal entries equal to one and all other entries equal to zero.
if A is an
matrix, then ImA = A = AIn.
3. The transpose is defined for nonsquare matrices, too.
Let
Matrix Multiplication
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Then
Theorem (Properties of the Transpose)
For matrices A and B of the right size and r a scalar,
• (AT)T = A
• (A + B)T = AT + BT
• (rA)T = rAT
• (AB)T = BTAT
Linear Combinations of Columns
Given a matrix
,
the columns of the matrix may be considered as
In other words
matrices themselves. Call these column matrices
A linear combination of columns is a matrix of the form
where
are scalars.
Example: Let
Transpose, Trace and Diagonal
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then
If c1 = 1,c2 = 2 and c3 = 3, then
and we say that
is a linear
combination of the columns of A.
Notice that
and, in general, for any choice of c1,c2,c3 we have
This illustrates an important principle that is valid for any matrix: B is linear combination of the columns
of a matrix A if and only if
Interesting additional note to the example: suppose we want to know if
columns of
is a linear combination of the
. In other words, can we find numbers c ,c ,c so that
Linear Combinations of Columns
1 2 3
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Notice that this is the same as solving the system of linear equations:
so we are just solving three equations in three unknowns.
The augmented matrix of this system of equations is
whose reduced row echelon form is
The last row says that there is no solution, so
is not a linear combination of the columns of A.
Now suppose we ask if
is a linear combination of the columns of the same matrix A. The augmented
matrix of the system of linear equations
which has an augmented matrix
Linear Combinations of Columns
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whose reduced row echelon form is
Hence we have a solution (in fact, an infinite number of them) so that
columns of A.
is a linear combination of the
Linear Combinations, Systems of Linear Equations and Matrix
Multiplication
We have been representing a system of m linear equations in n unknowns as
If A is the coefficient matrix, then these equations may be written in a matrix form:
We will usually write this equation as
where
and
.
We make two easy but important observations:
Linear Combinations, Systems of Linear Equations and Matrix Multiplication
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1.
if
2.
and
has a solution if and only if
are the columns of A, then
.
is a linear combination of the
.
Properties of Matrix Arithmetic
The following theorem proves five additive and five multiplicative properties of matrix addition and scalar
multiplication
Theorem (Properties of Addition and Scalar Multiplication)
Let A,B and C be
matrices, r and s be scalars. Then
A1: A + B is a matrix
A2: (A + B) + C = A + (B + C)
M1: rA is a matrix
M2: r(A + B) = rA + rB
A3: There exists a matrix
M3: (r + s)A = rA + sA
so that
A4: For every matrix A there exists a matrix − A so that
A5: A + B = B + A
Proofs: The proofs are based on the following known properties of real numbers:
M4: (rs)A = r(sA)
M5: 1A = A
• r + (s + t) = (r + s) + t (associative law of addition)
• r + s = s + r (commutative law of addition)
• r(st) = (rs)t (associative law of multiplication)
• rs = sr (commutative law of multiplication)
• r(s + t) = rs + rt (distributive law)
• r + 0 = r (0 is the additive identity)
• 1r = r (1 is the multiplicative identity)
Click on each statement to see its proof.
Theorem (Properties of Matrix Multiplication)
Let A, B and C be matrices of the right size for matrix multiplication. Then
• A(B + C) = AB + AC
• (B + C)A = BA + CA
• A(BC) = (AB)C
Properties of Matrix Arithmetic
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The zero matrix
The zero matrix
properties:
and the identity matrix In
is the
matrix with z = 0 for all i and j. It satisfies the following
i,j
Theorem (Properties of the Zero Matrix)
Suppose that A and
are of appropriate sizes in each case. Then
•
•
•
•
The identity matrix I = [a ] is the
(square) matrix where
Theorem (Properties
ofi,j the Identity Matrix)
n
Suppose that A is an
matrix. Then ImA = A = AIn.
Solving the Matrix Equation AX=B
Suppose that A is an
matrix and B is an
definition of matrix multiplication, X must be of size
then we have
matrix. We want find a matrix X so that AX = B. By the
Specifically, if A = [ai,j], X = [xi,j] and B = [bi,j],
If we look at the i-j entry on both sides of the equation, we get
If we keep j fixed and let i range from 1 to m, we get a system of linear equations
The zero matrix and the identity matrix In
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We can solve this system by finding the reduced row echelon form of the augmented matrix. Here's the beautiful
part: the n different systems of linear equations that arise as j takes on the values from 1 to n all have the identical
coefficient matrix. This means that the same sequence of elementary row operations may be used on each of the
equations to find the solutions.
Examples:
1. Let
and
. The X must be a
matrix so that
Solving for the first column means finding the reduced row echelon form of
which is
so that
,
,x =s
Solving for the second3,1column means finding the reduced row echelon form of
which is
Hence we conclude that
so that x
= − 1 + t, x
1,2
2,2
= 1 − 2t, x
=t
3,2
is a solution of the matrix equation AX = B for any choice of s and t.
Notice how similar are the two matrices to be put into reduced row echelon form. They differ, of
course, only in the last column. In fact, this means that exactly the same elementary row
operations were used to put both matrices into reduced row echelon form. Since both
computations used the same coefficient matrix, we could have carried out both computations at
once by starting with the matrix
and obtaining
for the reduced row echelon form.
2. An even more striking example can be obtained when there are no free variables. Let
Solving the Matrix Equation AX=B
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and
so that X is
Then the augmented
matrix
has reduced row echelon form
From the first column of B we get x1,1 = − 1, x2,1 = 5 and x3,1 = − 1.
From the second column of B we get x1,2 = − 1, x2,2 = 5 and x3,2 = − 1.
From the third column of B we get x1,3 = − 2, x2,3 = 7 and x3,3 = 1.
So
. Notice that X is the right half of the reduced row echelon
form, and note that it follows directly from the identity matrix in the left half of the same matrix.
3. If A and B are both square
matrices, and the reduced row echelon form of A is In, then the
reduced row echelon form of
[A | B] is [In | X] where X satisfies AX = B.
Solving the Matrix Equation AX=B
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