Matrices and Determinant: Types, order, Notation, basic operations, transpose, determinants of 2 x 2 and 3 x 3 matrices, Inverse of 2 x 2 matrix and application to simultaneous equation

SUBJECT: MATHEMATICS

CLASS: SS 3

TERM: FIRST TERM

WEEK 4                                               DATE…………………………….

MATRICES

*Definition of matrix and uses

*Examples and types of matrix

*Matrix addition and subtraction

*Multiplication of matrices

A matrix is an ordered set of numbers listed rectangular form. A matrix is, by definition, a rectangular array of numeric or algebraic quantities which are subject to mathematical operations. Matrices can be defined in terms of their dimensions (number of rows and columns). Let us take a look at a matrix with 4 rows and 3 columns (we denote it as a 4×3 matrix and call it A):

Each individual item in a matrix is called a cell, and can be denoted by the particular row and column it resides in. For instance, in matrix A, element a₃₂ can be found where the 3rd row and the 2nd column intersect. Matrices and Determinants were discovered and developed in the eighteenth and nineteenth centuries. Initially, their development dealt with transformation of geometric objects and solution of systems of linear equations. Historically, the early emphasis was on the determinant, not the matrix. In modern treatments of linear algebra, matrices are considered first. We will not speculate much on this issue.

Matrices provide a theoretically and practically useful way of approaching many types of problems including:

Here are a couple of examples of different types of matrices:

Symmetric

Diagonal

Upper Triangular

Lower Triangular

Zero

Identity

In addition to these basic types, there are many other specialized matrices that are used in different applications. For instance, a matrix can be used to represent transformations in computer graphics or information transmission through digital media. Matrices can also be used for solving systems of linear equations and for performing mathematical operations such as multiplication, addition, subtraction, and division.

One of the most important operations involving matrices is matrix multiplication. In order to multiply two matrices, their dimensions must be compatible. For instance, if we are multiplying a 2×3 matrix with a 3×2 matrix, the resulting product will be a 3×2 matrix. The process of multiplying matrices involves multiplying the corresponding cells in the matrix, and then summing these products together to obtain the final result.

There are also several different methods for performing matrix addition and subtraction. For instance, we could perform row-by-row addition of two matrices, or we could use a special type of algorithm called Gaussian elimination that is commonly used in linear algebra. Regardless of the approach used, matrix addition and subtraction can be a very useful tool for solving problems involving matrices.

And a fully expanded m×n matrix A, would look like this:

… or in a more compact form:

Example: Let A denote the matrix

        [2  5  7  8]
        [5  6  8  9]
        [3  9  0  1]

This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.We denote the element on the second row and fourth column with a_2,4.

Square matrix

If a matrix A has n rows and n columns then we say it’s a square matrix. In a square matrix the elements a_i,i , with i = 1,2,3,… , are called diagonal elements.

Remark: There is no difference between a 1 x 1 matrix and an ordinary number.

Diagonal matrix

A diagonal matrix is a square matrix with all de non-diagonal elements 0.

The diagonal matrix is completely defined by the diagonal elements.

Example:

 
        [7  0  0]
        [0  5  0]
        [0  0  6] The matrix is denoted by diagonal (7 , 5 , 6)

Row matrix :A matrix with one row is called a row matrix. [2 5 -1 5]

Column matrix:A matrix with one column is called a column matrix.

 
      [2]
      [4]
      [3]
      [0]

Matrices of the same kind:Matrix A and B are of the same kind if and only if A has as many rows as B and A has as many columns as B

        [7  1  2]       [4  0  3]
        [0  5  6]  and  [1  1  4]
        [3  4  6]       [8  6  2]

The transposed matrix of a matrix

The n x m matrix B is the transposed matrix of the m x n matrix A if and only if

The ith row of A = the ith column of B for (i = 1,2,3,..m)

So a_i,j = b_j,I  The transposed matrix of A is denoted T(A) or A^T

        [7  1 ]       [7  0  3]
        [0  5 ]    =  [1  5  4]
        [3  4 ]
 
0-matrix:When all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.
An identity matrix (I):An identity matrix I is a diagonal matrix with all the diagonal elements = I.

A scalar matrix S :A scalar matrix S is a diagonal matrix whose diagonal elements all contain the same scalar value.

a_1,1 = a_i,i for (i = 1,2,3,..n)

 
        [7  0  0]
        [0  7  0]
        [0  0  7]

The opposite matrix of a matrix: If we change the sign of all the elements of a matrix A, we have the opposite matrix -A.If A’ is the opposite of A then a_i,j‘ = -a_i,j, for all i and j.

A symmetric matrix:A square matrix is called symmetric if it is equal to its transpose.

Then a_i,j = a_j,i , for all i and j.

        [7 1 5]
        [1 3 0]
        [5 0 7]

A skew-symmetric matrix :A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.Then a_i,j = -a_j,i , for all i and j.

        [ 0  1 -5]
        [-1  0  0]
        [ 5  0  0]
 
 
 
 

Matrix Addition and Subtraction

DEFINITION: Two matrices A and B can be added or subtracted if and only if their dimensions are the same (i.e. both matrices have the same number of rows and columns.

Take:

… or in a more compact form:

Example: Let A denote the matrix

­­­­­­­ [[2 5 7 8]]

­ ­ ­ ­ ­ ​[[5 6 8 9]]

​ _​ [[3 9 0 1]] This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.

Adding two matrices:

If A and B are two matrices of the same kind, then adding A + B to both sides of an equation will produce a solution for any given equation as long as all terms on both sides of the equal sign are matrices and that we have the same kind of matrices on both sides.

Example

[[1 2]] + [[3 4]] = [[4 6]] ¯_____________________________________________________ Statement 1)

Addition

Addition and subtraction operations can easily be performed on matrices, provided the matrices have the same dimensions. All that is required is to add or subtract the corresponding cells of each matrix involved in the operation. Let us take a look at the addition of two 2×3 matrices, A and B:

If A and B above are matrices of the same type then the sum is found by adding the corresponding elements a_ij + b_ij .

Here is an example of adding A and B together.

Evaluation:

1. Evaluate | -5 0  |   | 6 -3 |
            |       | + |      |
            |_ 4 1 _|   |_2  3_|

Subtraction of matrices is done in the same manner as addition. Always be aware of the negative signs and remember that a double negative is a positive!

SUBTRACTION

If A and B are matrices of the same type then the subtraction is found by subtracting the corresponding elements a_ij − b_ij.

Here is an example of subtracting matrices.

Example: Consider the three matrices J, F, and M from above. Evaluate

Answer.

We have

and since

we get

To compute J–M, we note first that

Since J-M = J + (-1)M, we get

And finally, for J-F+2M, we have a choice. Here we would like to emphasize the fact that addition of matrices may involve more than one matrix. In this case, you may perform the calculations in any order. This is called associativity of the operations. So first we will take care of -F and 2M to get

Since J-F+2M = J + (-1)F + 2M, we get

So first we will evaluate J-F to get

to which we add 2M, to finally obtain

Evaluation: Given    evaluate: 5A – 2B

MULTIPLICATION

Performing the operation product involves multiplying the cells of a particular rows in the first matrix by the cells of a particular column in the second matrix, adding the products, and storing the result in the cell of the resultant matrix whose coordinates correspond to the row of the first matrix and the column of the second matrix. For instance, in AB = C, if we want to find the value of c12, we must multiply the cells of row 1 in the first matrix by the cells of column 2 in the second matrix and sum the results.

Example 1

Multiply:

Answer

This is 2×3 times 3×2, which will give us a 2×2 answer.

Our answer is a 2×2 matrix

Multiplying 2 × 2 Matrices

The process is the same for any size matrix. We multiply across rows of the first matrix and down columns of the second matrix, element by element. We then add the products:

In this case, we multiply a 2 × 2 matrix by a 2 × 2 matrix and we get a 2 × 2 matrix as the result.

Example 2:

Multiply:

Answer

Note 2 : Commutativity of Matrix Multiplication:

Does AB = BA?

Let’s see if it is true using an example

Example 3:

If

and

find AB and BA.

Answer

We performed AB above, and the answer was:

Now BA is (3 × 2)(2 × 3) which will give 3 × 3:

So in this case, AB does NOT equal BA

In general, when multiplying matrices, the commutative law doesn’t hold, i.e. AB ≠ BA. There are two common exceptions to this:

• The identity matrix: IA = AI = A.
• The inverse of a matrix: A^-1A = AA^-1 = I.

Example 4 – Multiplying by the Identity Matrix

Given that

find AI.

Answer

We see that multiplying by the identity matrix does not change the value of the original matrix.That is, AI = A

Further Examples

Example1. If possible, find BA and AB.

Answer

AB is not possible. (3 × 3) × (1 × 3).

Example2. Determine if B = A^-1.

Answer

If B = A^-1, then AB = I.

So B is NOT the inverse of A.

Example3. In studying the motion of electrons, one of the Pauli spin matrices is

where

Show that s² = I. [If you have never seen j before, it is on complex numbers].

Answer

Example4. Evaluate the following matrix multiplication which is used in directing the motion of a robotic mechanism.

Answer

The interpretation of this is that the robot arm moves from position (2, 4, 0) to position (-2.46, 3.73, 0). That is, it moves in the x-y plane, but its height remains at z = 0. The 3 × 3 matrix containing sin and cos values tells it how many degrees to move.

 TRANSPOSE AND INVERSE OF MATRICES, DETERMINANT OF MATRICES, APPLICATION OF DETERMINANTS

TRANSPOSE OF MATRICES

DEFINITION: The transpose of a matrix is found by exchanging rows for columns i.e. Matrix A = (a_ij) and the transpose of A is:

A^T = (a_ji) where j is the column number and i is the row number of matrix A.

For example, the transpose of a matrix would be:

In the case of a square matrix (m = n), the transpose can be used to check if a matrix is symmetric. For a symmetric matrix A = A^T.

If, in addition to being square, all the elements are real numbers, then A is called a real symmetric matrix. Its transpose will be its inverse (see below).

INVERSE OF MATRICES

DEFINITION: The inverse of matrix A can be found using the following formula:

A-1 = adjoint/adj(A)

Where A is a square matrix, we can say that:

1. If A is a real symmetric matrix, then its inverse will be its transpose.

2. If A is a positive definite (all non-zero elements are positive) matrix, then it has an inverse and its inverse is also a positive definite matrix.

3. The determinant of the inverse will be the reciprocal of its determinant (Reciprocal of A = adj(A)-1).

The Determinant of a Matrix

DEFINITION: Determinants play an important role in finding the inverse of a matrix and also in solving systems of linear equations. In the following we assume we have a square matrix (m = n). The determinant of a matrix A will be denoted by det(A) or |A|. Firstly the determinant of a 2×2 and 3×3 matrix will be introduced, then the n×n case will be shown.

Determinant of a 2×2 matrix

Assuming A is an arbitrary 2×2 matrix A, where the elements are given by:

then the determinant of a this matrix is as follows:

Determinant of a 3×3 matrix

The determinant of a 3×3 matrix is a little more tricky and is found as follows (for this case assume A is an arbitrary 3×3 matrix A, where the elements are given below).

then the determinant of a this matrix is as follows:

Determinant of a n×n matrix

For the general case, where A is an n×n matrix the determinant is given by:

Where the coefficients α_ij are given by the relation:

where β_ij is the determinant of the (n-1) × (n-1) matrix that is obtained by deleting row i and column j. This coefficient α_ij is also called the cofactor of a_ij.

Calculating a 2 × 2 Determinant

In general, we find the value of a 2 × 2 determinant with elements a, b, c, d as follows:

We multiply the diagonals (top left × bottom right first), then subtract.

Example 1

The final result is a single number.

Application of Determinants in Solve Systems of Equations

We can solve a system of equations using determinants, but it becomes very tedious for large systems. We will only do 2 × 2 and 3 × 3 systems using determinants.

Crammer’s Rule

The solution (x, y) of the system

can be found using determinants:

Example 2

Solve the system using Crammer’s Rule:x − 3y = 6 ; 2x + 3y = 3

Answer

First we determine the values we will need for Cramer’s Rule:

a₁ = 1 b₁ = -3 c₁ = 6

a₂ = 2 b₂ = 3 c₂ = 3

3 × 3 Determinants

A 3 × 3 determinant

can be evaluated in various ways.

We will use the method called “expansion by minors”. But first, we need a definition.

Cofactors

The 2 × 2 determinant

is called the cofactor of a₁ for the 3 × 3 determinant:

The cofactor is formed from the elements that are not in the same row as a₁ and not in the same column as a₁.

Similarly, the determinant

is called the cofactor of a₂. It is formed from the elements not in the same row as a₂and not in the same column as a₂.

We continue the pattern for the cofactor of a₃.

Expansion by Minors

We evaluate our 3 × 3 determinant using expansion by minors. This involves multiplying the elements in the first column of the determinant by the cofactors of those elements. We subtract the middle product and add the final product.

Note that we are working down the first column and multiplying by the cofactor of each element.

Example 3

Evaluate

Answer

= -2[(-1)(2) − (-8)(4)] − 5[(3)(2) − (-8)(-1)] + 4[(3)(4) − (-1)(-1)]

= -2(30) − 5(-2) + 4(11)

= -60 + 10 + 44= -6

Here, we are expanding by the first column. We can do the expansion by using the first row and we will get the same result.

Crammer’s Rule to Solve 3 × 3 Systems of Linear Equations  

We can solve the general system of equations,

a₁x + b₁y + c₁z = d₁

a₂x + b₂y + c₂z = d₂

a₃x + b₃y + c₃z = d₃

By using the determinants:

where

Example 4

Solve, using Crammer’s Rule:

2x + 3y + z = 2

−x + 2y + 3z = −1

−3x − 3y + z = 0

Answer

where

So

Determinant Exercises

Evaluation

1. Evaluate by expansion of minors:

2. Solve the system by use of determinants:

x + 3y + z = 4

2x − 6y − 3z = 10

4x − 9y + 3z = 4

The Inverse of a Matrix

DEFINITION: Assuming we have a square matrix A, which is non-singular (i.e. det(A) does not equal zero), then there exists an n×n matrix A^-1 which is called the inverse of A, such that this property holds:

AA^-1 = A^-1A = I, where I is the identity matrix.

The inverse of a 2×2 matrix

Take for example an arbitrary 2×2 Matrix A whose determinant (ad − bc) is not equal to zero.

where a,b,c,d are numbers, The inverse is:

Now try finding the inverse of your own 2×2 matrices.

The inverse of a n×n matrix: The inverse of a general n×n matrix A can be found by using the following equation.

Where the adj(A) denotes the adjoint (or adjugate) of a matrix. It can be calculated by the following method:

• Given the n×n matrix A, define
• B = b_ij
• to be the matrix whose coefficients are found by taking the determinant of the (n-1) × (n-1) matrix obtained by deleting the i^th row and j^th column of A. The terms of B (i.e. B = b_ij) are known as the cofactors of A.
• Define the matrix C, where
• c_ij = (−1)^i+j b_ij.
• The transpose of C (i.e. C^T) is called the adjoint of matrix A.

Lastly to find the inverse of A divide the matrix C^T by the determinant of A to give its inverse.

We’ll find the inverse of a matrix using 2 different methods. You can decide which one to use depending on the situation.

The first method is limited to finding the inverse of 2 × 2 matrices. It involves the use of the determinant of a matrix which we saw earlier.

Reminder: We can only find the determinant of a square matrix. For example, if A is the square matrix

Then we can find the determinant of A:

= 10 + 3 = 13.

For convenience, we could have written the determinant of A as |A| and so our final answer would be:

|A| = 13

Another way of writing the same thing is to use “det” for “determinant”. So for example, in this case we would write: det(A) = 13

Method 1 – Transposing and Determinants

This method is only good for finding the inverse of a 2 × 2 matrix.

Example.

Find the inverse, A^-1, of     using Method 1.

Answer

Method 1 is as follows. [1] Interchange leading diagonal elements:

-7 → 2; 2 → -7

[2] Change signs of the other 2 elements:

-3 → 3; 4 → -4

[3] Find the determinant |A|

= -14 + 12 = -2

Multiply result of [2] by

So we have found the inverse, as required.

Is it correct?

We check by multiplying our inverse by the original matrix. If we get the identity matrix (I) for our answer, then we must have the correct answer.

Method 2: Adjoint matrix

Method 2 uses the adjoint matrix method.

Answer

The inverse of a 3×3 matrix is given by:

“adj A” is short for “the adjoint of A“. We use cofactors (that we met earlier) to determine the adjoint of a matrix.

Cofactors  Recall: The cofactor of an element in a matrix is the value obtained by evaluating the determinant formed by the elements not in that particular row or column.

Example: Consider the matrix

The cofactor of 6 is

The cofactor of -3 is

We find the adjoint matrix by replacing each element in the matrix with its cofactor and applying a + or – sign as follows:

and then finding the transpose of the resulting matrix. The transpose means the 1^stcolumn becomes the 1^st row; 2^nd column becomes 2^nd row, etc.

Example 1:

Find the inverse of the following by using the adjoint matrix method:

A = 

Solution:

Step 1:

Replace elements with cofactors and apply + and –

Step 2

Transpose the matrix:

adjA =

Before we can find the inverse of matrix A, we need det A:

Now we have what we need to apply the formula

So,

A^-1 =

Example 2.

Find the inverse of

using method 2.

                                         Answer

Interchange rows and columns:

Det A =       So

General Evaluation

1.   Solve for x and y , ……..x= ,y=

2.   If X =  and Y =  , Find XY …………

3.    Determine X + Y if 

4.   If , find u and v if x = 3, y = 1 and A =30⁰

Reading Assignment: NGM for SS 3 Chapter 8 page 64

Weekend Assignment

1.   Find the non-zero positive value of x which satisfies the equation    A. 2 B   C.   D. 1

2.   Find the value of k.  , A 1 B 2 C 3 D 4

3.   Find the matrix T if ST = I where S =  and I is the identity matrix. ………

4.   Given that Q =  , evaluate

5.   A matrix P =  is such that P^T = -P , where P^T is the transpose of P . I f b = I, then P is. ………….

6. Given A = ​ and B = , find AB. ………

7. Find the inverse of M if ImM=A where A is as given below:

1 -1 2 0 1

-2 4 3 5 6

8. Write an expression that represents the determinant of a 2×2 matrix. …………..

9. Give 3 examples of the use of inverse matrices

10. Is it possible to find A-1 without finding det(A)? Justify your answer with an example

THEORY

1.   Find the inverse of the matrix

2.   Find the values of t for which the determinant of the matrix below will give zero.