Properties of Matrices
These are my notes on properties of matrices.
Matrices occur in many fields of study and have a wide variety of applications.
The following notation is used to denote elements in a matrix.
\(\begin{pmatrix}
a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} \end{pmatrix}\)
A general element of the matrix is denoted by \(a_{ij}\). This refers to the
element in the ith row and jth column. Two m by n matrices are equal if
corresponding elements are equal. If a and B have different dimensions, they
cannot be equal.
Example 1
Determining matrix elements
Let \(a_{ij}\) denote a general element in A and \(b_{ij}\) a general element in
B, where:
\(A =
\begin{pmatrix}3&-3&7\\1&6&-2\\4&2&5\end{pmatrix}.....\begin{pmatrix}3&x&7\\1&6&-2\\4&5&2\end{pmatrix}\)
Identify \(a_{12}\), \(b_{32}\), and \(a_{13}\)
Compute \(a_{31}b_{13} + a_{32}b_{23} + a_{33}b_{33}\)
Is there a value for x that will make the statement A=B true?
Solution:
The element \(a_{12}\) is located in the first row, second column of A. So,
\(a_{12}=-3\). In a similar manner, \(b_{32}=5\) and \(a_{13}=7\).
\(a_{31}b_{13} + a_{32}b_{23} + a_{33}b_{33} = (4)(7) + (2)(-2) + (5)(2) = 34\)
No, since \(a_{32} = 2 \not = 5 = b_{32}\) and \(a_{33} = 5 \not = 2 = b_{33}\).
Even if we let x=-3, there are other corresponding elements in A and B that are
not equal.
Example 2
Applying addition of matrices
Increase the contrast of the + sign by changing light gray and dark gray to
black. Uses matrices to represent this computation.
Solution:
We have a matrix:
\(\begin{pmatrix} 1&2&1\\2&2&2\\1&2&1 \end{pmatrix}\)
To change the contrast, we can reduce each 1 in the matrix to 0 and increase
each 2 to 3. The addition of the matrix B can accomplish this task.
\(A+B=\begin{pmatrix}1&2&1\\2&2&2\\1&2&1\end{pmatrix}+\begin{pmatrix}1&1&-1\\1&1&1\\1&1-1\end{pmatrix}=\begin{pmatrix}0&3&0\\3&3&3\\0&3&0\end{pmatrix}\)
Example 3
Add matrix A to matrix B
A=\(\begin{pmatrix}7&8&-1\\0&-1&6\end{pmatrix}\)
B=\(\begin{pmatrix}5&-2&10\\3&2&4\end{pmatrix}\)
Solution:
Adding is straightforward, look at each individual element.
\(\begin{pmatrix}12&6&9\\-3&1&10\end{pmatrix}\)
Example 4
Add matrix B to A
Solution:
These are the same matrices as above.
Adding is again straightforward, take each element one by one.
\(\begin{pmatrix}12&6&9\\-3&1&10\end{pmatrix}\)
Notice that A+B=B+A. The commutative property for matrix addition holds in
general, provided that A and B have the same dimensions.
Example 5
Substract: A-B
Solution:
We are using the same matrices, A and B
As before, take each element turn by turn
\(\begin{pmatrix}2&10&-11\\3&-3&2\end{pmatrix}\)
Multiplying a matrix by a scaler
A simple way to write the expression A+B+B is A + 2B. Multiplying B by 2 to
obtain 2B is called scalar multiplication. Each element of B will be multiplied
by 2.
Sometimes a matrix B is denoted \(B=[b_{ij}]\), where \(B_{ij}\) represents the
element in the ith row, jth column. In this way, we could write 2B as
\(2[b_{ij}]=[2B_{ij}]\). This indicates that to calculate 2B, multiply each
\(b_{ij}\) by 2.
Example 6
Performing scalar multiplication
If \(A=\begin{pmatrix}2&7&11\\-1&3&-5\\0&9&-12\end{pmatrix}\), find -4A
Solution:
Take each element in turn and multiply each by -4
\(\begin{pmatrix}-8&-28&-44\\4&-12&20\\0&-36&48\end{pmatrix}\)
Example 7
Performing operations on matrices.
If possible, perform the indicated operation.
A+3B;A-C;-2C-3D
A=\(\begin{pmatrix}4&-2\\3&5\end{pmatrix}\)
B=\(\begin{pmatrix}0&1\\2&3\end{pmatrix}\)
C=\(\begin{pmatrix}1&-1\\0&7\\-4&2\end{pmatrix}\)
D=\(\begin{pmatrix}-1&-3\\9&-7\\1&8\end{pmatrix}\)
Solution:
A=\(\begin{pmatrix}4&1\\-3&14\end{pmatrix}\)
B=A-C is undefined because the dimensions are different of the two matrices
involved.
C= \(\begin{pmatrix}1&11\\-27&7\\5&-28\end{pmatrix}\)
Matrix Products
Addition, subtraction, and multiplication can be performed on numbers,
variables, and functions. The same operations apply to matrices. matrix
multiplication is different than scalar multiplication.
The product of an m*n matrix and an n*k matrix is the m*k matrix AB, which is
computed as follows. To find the element of AB in the ith row and jth column,
multiply each element of b. The sum of these products will give the element of
row 1 and column j in AB.
In order to compute the product of two matrices, the number of columns in the
first matrix must equal the number of rows in the second matrix.
Example 8
Multiplying Matrices
Compute the product of:
\(\begin{pmatrix}1&-1\\0&3\\4&-2\end{pmatrix}\) \(\begin{pmatrix}-1\\2\end{pmatrix}\)
Solution:
The dimension of A is 3*2 and the dimension of b is 2*1. The dimension of AB
will be 3*1 and can be found like this:
\(AB=\begin{pmatrix}1&-1\\0&3\\4&-2\end{pmatrix}*\begin{pmatrix}-1\\-2\end{pmatrix}=\begin{pmatrix}(1)(-1)+(-1)(-2)\\(0)(-1)+(3)(-2)\\(4)(-1)+(-2)(-2)\end{pmatrix}=\begin{pmatrix}1\\-6\\0\end{pmatrix}\)
With matrix multiplication, it is shown that \(AB\not=BA\). Unlike
multiplication of numbers, variables, and functions, matrix multiplication is
not commutative. Instead, matrix multiplication is similar to function
composition.
Square matrices have the same number of rows as columns and have dimension n*n
for some natural number n. When we multiply two square matrices, both having
dimension n*n, the resulting matrix also had dimension n*n.
Example 9
Multiplying Square Matrices
\(A=\begin{pmatrix}1&0&7\\3&2&-1\\-5&-2&5\end{pmatrix}-B=\begin{pmatrix}4&-6&7\\8&9&10\\0&1&-3\end{pmatrix}\)
Solution:
\(AB=\begin{pmatrix}1(4)+0(8)+7(0)|1(-6)+0(9)+7(1)|1(7)+0(10)+7(-3)\\
3(4)+2(8)-1(0)|3(-6)+2(9)-1(1)|3(7)+2(10)-1(-3)\\
-5(4)-2(8)+5(0)|-5(-6)-2(9)+5(1)|-5(7)-2(10)+5(-3)\end{pmatrix}\)
\(\begin{pmatrix}4&1&-14\\28&-1&44\\-36&17&-70\end{pmatrix}\)
Real numbers satisfy the commutative, associative, and distributive properties
for various arithmetic operations. matrices also satisfy soem of these
properties, provided that their dimensions are valid so that the resulting
expressions are defined.