Mathematics: Linear Algebra

Matrix Basics
A matrix (plural: matrices) is a rectangular array of numbers or variables, often called elements, which are arranged in columns and rows. A matrix is generally represented by a capital letter, with its elements represented by the corresponding lowercase letter with two subscripts indicating the row and column of the element. For example,
 represents the element in row a column b of matrix N.


A matrix can be described in terms of the number of rows and columns it contains in the format  is the number of rows and <i>b</i> is the number of columns. The matrix shown above is a <br><img data-cke-saved-src=" /> matrix. Any
 matrix where  is a square matrix. A vector is a matrix that has exactly one column (column vector) or exactly one row (row vector).

The main diagonal of a matrix is the set of elements on the diagonal from the top left to the bottom right of a matrix. Because of the way it is defined, only square matrices will have a main diagonal. For the matrix shown below, the main diagonal consists of the elements
.


A  matrix such as the one shown below would not have a main diagonal because there is no straight line of elements between the top left corner and the bottom right corner that joins the elements.


A diagonal matrix is a square matrix that has a zero for every element in the matrix except the elements on the main diagonal. All the elements on the main diagonal must be nonzero numbers.


If every element on the main diagonal of a diagonal matrix is equal to one, the matrix is called an identity matrix. The identity matrix is often represented by the letter I.


A zero matrix is a matrix that has zero as the value for every element in the matrix.


The zero matrix is the identity for matrix addition. Do not confuse the zero matrix with the identity matrix.

The negative of a matrix is also known as the additive inverse of a matrix. If matrix N is the given matrix, then matrix –N is its negative. This means that every element  is equal to  in the negative. To find the negative of a given matrix, change the sign of every element in the matrix and keep all elements in their original corresponding positions in the matrix.

If two matrices have the same order and all corresponding elements in the two matrices are the same, then the two matrices are equal matrices.

A matrix N may be transposed to matrix  by changing all rows into columns and changing all columns into rows. The easiest way to accomplish this is to swap the positions of the row and column notations for each element. For example, suppose the element in the second row of the third column of matrix N is
. In the transposed matrix
, the transposed element would be
, and it would be placed in the third row of the second column.


To quickly transpose a matrix by hand, begin with the first column and rewrite a new matrix with those same elements in the same order in the first row. Write the elements from the second column of the original matrix in the second row of the transposed matrix. Continue this process until all columns have been completed. If the original matrix is identical to the transposed matrix, the matrices are symmetric.

The determinant of a matrix is a scalar value that is calculated by taking into account all the elements of a square matrix. A determinant only exists for square matrices. Finding the determinant of a  matrix is as simple as remembering a simple equation. For a
 matrix
, the determinant is obtained by the equation
. Anything larger than
 requires multiple steps. Take matrix
. The determinant of N is calculated as
 or
.

There is a shortcut for  matrices: add the products of each unique set of elements diagonally left-to-right and subtract the products of each unique set of elements diagonally right-to-left.

In matrix N, the left-to-right diagonal elements are
,
, and
.
The right-to-left diagonal elements are ,
, and .
.
Calculating the determinants of matrices larger than
 is rarely, if ever, done by hand.

The inverse of a matrix M is the matrix that, when multiplied by matrix M, yields a product that is the identity matrix. Multiplication of matrices will be explained in greater detail shortly. Not all matrices have inverses. Only a square matrix whose determinant is not zero has an inverse. If a matrix has an inverse, that inverse is unique to that matrix. For any matrix M that has an inverse, the inverse is represented by the symbol
. To calculate the inverse of a  square matrix, use the following pattern:


Another way to find the inverse of a matrix by hand is use an augmented matrix and elementary row operations. An augmented matrix is formed by appending the entries from one matrix onto the end of another.

For example, given a  invertible matrix
, you can find the inverse
 by creating an augmented matrix by appending a  identity matrix:
. To find the inverse of the original
 matrix, perform elementary row operations to convert the original matrix on the left to an identity matrix:
. For instance, the first step might be to multiply the second row by
 and then subtract it from the first row to make its second column a zero. The end result is that the
 section on the right will become the inverse of the original matrix:
.

Elementary row operations
Elementary row operations include multiplying a row by a non-zero scalar, adding scalar multiples of two rows, and switching rows. These operations can be done using matrix multiplication with specialized transformation matrices. Row switching is achieved by swapping the corresponding rows in the identity matrix. For example, consider switching row
2 and row 3 in a  matrix:


The transformation matrix for row multiplication is also based on the identity matrix with the scalar multiplication factor in place of the 1 in the corresponding row. Multiplying row 1 by  in a
 matrix:


The transformation matrix for row addition consists of the identity matrix with a 1 in the element corresponding to the two rows being added in the column where you want the result to go. Adding row 2 to row 1 in a  matrix:


Basic Operations with Matrices
There are two categories of basic operations with regard to matrices: operations between a matrix and a scalar, and operations between two matrices.

Scalar Operations
A scalar being added to a matrix is treated as though it were being added to each element of the matrix:


The same is true for the other three operations.
Subtraction:

Multiplication:

Division:


Matrix Addition and Subtraction
All four of the basic operations can be used with operations between matrices (although division is usually discarded in favor of multiplication by the inverse), but there are restrictions on the situations in which they can be used. Matrices that meet all the qualifications for a given operation are called conformable matrices. However, conformability is specific to the operation; two matrices that are conformable for addition are not necessarily conformable for multiplication.

For two matrices to be conformable for addition or subtraction, they must be of the same dimension; otherwise the operation is not defined. If matrix M is a
 matrix and matrix N is a
 matrix, the operations
 and
 are meaningless.

If matrices M and N are the same size, the operation is as simple as adding or subtracting all of the corresponding elements:



The result of addition or subtraction is a matrix of the same dimension as the two original matrices involved in the operation.

Matrix Multiplication
The first thing it is necessary to understand about matrix multiplication is that it is not commutative. In scalar multiplication, the operation is commutative, meaning that
.
For matrix multiplication, this is not the case:
. The terminology must be specific when describing matrix multiplication. The operation
 can be described as A multiplied (or post-multiplied) by B, or B
pre-multiplied by A.

For two matrices to be conformable for multiplication, they need not be of the same dimension, but specific dimensions must correspond.

Taking the example of two matrices M and N to be multiplied
must equal <i>c</i> if the two matrices are to be conformable for this multiplication. The matrix that results from the multiplication will have the dimensions <br><img src=" />a and d are both equal to 1, the product is simply a scalar. Square matrices of the same dimensions are always conformable for multiplication, and their product is always a matrix of the same size.

The simplest type of matrix multiplication is a
 matrix (a row vector) times a
 matrix (a column vector). These will multiply in the following way:


The two matrices are conformable for multiplication because matrix M has the same number of columns as matrix N has rows. Because the other dimensions are both 1, the result is a scalar. Expanding our matrices to
 and , the process is the same:


Once again, the result is a scalar. This type of basic matrix multiplication is the building block for the multiplication of larger matrices.

To multiply larger matrices, treat each row from the first matrix and each column from the second matrix as individual vectors and follow the pattern for multiplying vectors. The scalar value found from multiplying the first-row vector by the first column vector is placed in the first row, first column of the new matrix. The scalar value found from multiplying the second-row vector by the first column vector is placed in the second row, first column of the new matrix. Continue this pattern until each row of the first matrix has been multiplied by each column of the second vector.

Below is an example of the multiplication of a  matrix and a  matrix.


This process starts by taking the first column of the second matrix and running it through each row of the first matrix. Removing all but the first M row and first N column, we would see only the following:


The first product would then be . This process will be continued for each column of the N matrix to find the first full row of the product matrix, as shown below.


After completing the first row, the next step would be to simply move to the second row of the M matrix and repeat the process until all of the rows have been finished. The result is a  matrix.


If the operation were done in reverse
, the result would be a
 matrix.


Matrices can be used to represent the coefficients of a system of linear equations and can be very useful in solving those systems. Take for instance three equations with three variables where all a, b, c, and d are known constants:






To solve this system, define three matrices:


The three equations in our system can be fully represented by a single matrix equation:


We know that the identity matrix times X is equal to X, and we know that any matrix multiplied by its inverse is equal to the identity matrix.


Our goal then is to find the inverse of A, or A-1. Once we have that, we can pre-multiply matrix D by A-1 (post-multiplying here is an undefined operation) to find matrix X.

Systems of equations can also be solved using the transformation of an augmented matrix in a process similar to that for finding a matrix inverse. Begin by arranging each equation of the system in the following format:



Define matrices A and D and combine them into augmented matrix Aa:

To solve the augmented matrix and the system of equations, use elementary row operations to form an identity matrix in the first
 section. When this is complete, the values in the last column are the solutions to the system of equations:


If an identity matrix is not possible, the system of equations has no unique solution. Sometimes only a partial solution will be possible. The following are partial solutions you may find:
 gives the non-unique solution

 gives the non-unique solution


This process can be used to solve systems of equations with any number of variables, but three is the upper limit for practical purposes.
Anything more ought to be done with a graphing calculator.

Reduced Row-Echelon Forms
When a system of equations has a solution, finding the transformation of the augmented matrix will result in one of three reduced row-echelon forms.  Only one of these forms will give a unique solution to the system of equations, however.  The following examples show the solutions indicated by particular results:
 gives the unique solution

 gives a
 gives a

Geometric Transformations
The four geometric transformations are translations, reflections, rotations, and dilations. When geometric transformations are expressed as matrices, the process of performing the transformations is simplified. For calculations of the geometric transformations of a planar figure, make a
 matrix, where n is the number of vertices in the planar figure. Each column represents the rectangular coordinates of one vertex of the figure, with the top row containing the values of the x-coordinates and the bottom row containing the values of the y-coordinates. For example, given a planar triangular figure with coordinates
,
, and , the corresponding matrix is
. You can then perform the necessary transformations on this matrix to determine the coordinates of the resulting figure.

Translation
A translation moves a figure along the x-axis, the y-axis, or both axes without changing the size or shape of the figure. To calculate the new coordinates of a planar figure following a translation, set up a matrix of the coordinates and a matrix of the translation values and add the two matrices.

where h is the number of units the figure is moved along the x-axis (horizontally) and v is the number of units the figure is moved along the y-axis (vertically).

Reflection
To find the reflection of a planar figure over the x-axis, set up a matrix of the coordinates of the vertices and pre-multiply the matrix by the
 matrix
 so that
. To find the reflection of a planar figure over the y-axis, set up a matrix of the coordinates of the vertices and pre-multiply the matrix by the  matrix
 so that
. To find the reflection of a planar figure over the line , set up a matrix of the coordinates of the vertices and pre-multiply the matrix by the  matrix
 so that
. Remember that the order of multiplication is important when multiplying matrices. The commutative property does not apply.

Rotation
To find the coordinates of the figure formed by rotating a planar figure about the origin θ degrees in a counterclockwise direction, set up a matrix of the coordinates of the vertices and pre-multiply the matrix by the
 matrix
.

For example, if you want to rotate a figure 90° clockwise around the origin, you would have to convert the degree measure to 270° counterclockwise and solve the
 matrix you have set as the pre-multiplier:
. Use this as the pre-multiplier for the matrix
 and solve to find the new coordinates.

Dilation
To find the dilation of a planar figure by a scale factor of k, set up a matrix of the coordinates of the vertices of the planar figure and pre-multiply the matrix by the
 matrix
 so that
.

This is effectively the same as multiplying the matrix by the scalar k, but the matrix equation would still be necessary if the figure were being dilated by different factors in vertical and horizontal directions. The scale factor k will be greater than 1 if the figure is being enlarged, and between 0 and 1 if the figure is being shrunk. Again, remember that when multiplying matrices, the order of the matrices is important. The commutative property does not apply, and the matrix with the coordinates of the figure must be the second matrix.
 

Problems:
P1. A sporting-goods store sells baseballs, volleyballs, and basketballs.

Baseballs $3 each
Volleyballs $8 each
Basketballs $15 each

Here are the same store's sales numbers for one weekend:



Find the total sales for each day by multiplying matrices.

P2. Given the following matrices, perform the operations if possible:

(a)

(b)

(c)

(d)

(e)

(f)


P3. Solve the following system of equations using an augmented matrix and elementary row operations:




 

Solutions:

P1. The first table can be represented by the following column-vector:

And the second table can be represented by this matrix:

Multiplying the second matrix by the first will result in a column vector showing the total sales for each day:

From this, we can see that Friday's sales were $107, Saturday's sales were $195, and Sunday's sales were $126.

P2. (a)

(b)

(c)


(d)


The determinant is not defined for a non-square matrix.

(e)

(f)

Recall that for matrix multiplication the order of terms matters, but not the grouping. That means
 or any grouping that maintains the order will give the same result.

P3. Begin by setting up the augmented matrix:

Using elementary row operations, there are many ways to arrive at the answer. Here is one way (note that the row operations refer to the values in the row from the previous matrix):



This reduced row echelon form indicates that the values that satisfy the original system of equations are
.