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Operations with Matrices

Operations with Matrices

How would you like to know the interaction between the organized arrangement of numbers? The matrix (not the movie) shall be discussed hereafter to give you a comprehensive insight into the interaction of the organized arrangement of numbers.

What does operations with matrices mean?

Operations with matrices comprise all calculations which are carried out with matrices. This could range from addition, multiplication, subtraction, a mix of any, transposition etc.

What is a matrix?

A matrix is just a rectangular or square collection of numbers that have been arranged in rows and columns.

What is an element?

An element is an individual number or variable which is arranged in a matrix. For instance in the 2 by 2 matrix ; 2, 3, a and b are all elements of matrix A.

Furthermore, elements of a matrix are arranged in arrays. The knowledge of matrices are used in calculating vectors and their operations, geometry as well as linear equations. Also, a matrix is classified by how many rows and columns it has. This eventually determines its name. To illustrate better, matrix has three rows (i.e. 1, 2 ; 3, 4 ; 5, 6 ) and two columns (i.e. 1, 3, 5 ; 2, 4, 6 ) so we call it 3 by 2 or 3 × 2 matrix. So, the idea of naming comes from the "row by column", just remember that phrase and you would know the matrix type.

Basic operations with matrices

Primary calculations in matrices are regarded as basic operations in matrices. Shortly, you shall be learning about them.

Addition of matrices

It is interesting to know that two or more matrices can be summed up to become one matrix. Although, for this to occur, these matrices must be of the same type or dimension. This means that on one hand a 2 by 2 matrix only gets added to a 2 by 2 matrix; and on the other hand, a 2 by 3 matrix cannot be summed with a 3 by 2 matrix.

When adding matrices of the same dimension, only elements occupying similar positions in their matrix are summed. This means that for a matrix and another matrix

You should always note that addition in matrices considers position firstly before the sum of the elements takes place.

Find P + Q if and

Solution:

Subtraction of matrices

The difference between two or more matrices can be calculated. Although, for this to occur, these matrices have the same dimension. This means that on one hand a 2 by 2 matrix only gets subtracted from a 2 by 2 matrix; and on the other hand, a 2 by 3 matrix cannot be subtracted from a 3 by 2 matrix.

When subtracting matrices of the same dimension, only elements occupying similar positions in their matrix are subtracted. Also, a difference between the two matrices would involve the subtraction of the second matrix from the first matrix. This means that for a matrix and another matrix

Then

You should always note that the difference of matrices considers position before the subtraction of the elements takes place.

Find P - Q ifand

Solution:

Combined or mixed matrix operations

Combined or mixed matrix operations are problems that require the application of more than one basic matrix operation.

Calculate 2B - A for

Solution:

Operation on matrices types

There are several kinds of operations that are carried out with matrices. The most fundamental of them is regarded as basic while a combination of these operations is regarded as mixed, and these have just been discussed. However, there are several other operations that are going to be explained hereafter.

Multiplication of a matrix by a scalar

This is also known as multiplication by a constant. Here, the constant is used to multiply each element of a matrix. If 2 is a constant used to multiply matrix, then

Negative of a matrix

Using the knowledge of multiplying a matrix by a scalar, it is possible to get the negative of a matrix. This is simply done by multiplying each element of a matrix by -1. Note that signs change depending on the sign any element of the matrix possesses. Therefore the negative of a matrix becomes

You should know that this principle can be applied if a matrix is to be divided by a constant.

Multiplication of matrices

The product of matrices can be obtained, but in order to do that, one condition must be met. The number of columns of the first matrix must be equal to the number of rows in the second matrix. So it means that a 2 by 3 matrix can multiply a 3 by 1 matrix because the first matrix has 3 columns while the second has 3 rows.

However, a 2 by 3 matrix cannot multiply a 2 by 2 matrix because the first matrix has 3 columns while the second matrix has 2 rows. This makes the operation between a 2 by 3 matrix and a 2 by 2 matrix not commutable which means it cannot be calculated.

For ease in representing matrices, we shall be writing each unit matrix with its dimension. So, for a matrix G which is a 3 by 2 matrix, we will write it as G3×2. This will help us in writing out matrices easily.

Determining matrix dimensions after multiplication

One question that may linger in your mind before and during the multiplication of matrices is; what dimension is the new matrix meant to have? Knowing the dimension of the resultant matrix gives you confidence and serves as a guide while multiplying matrices.

Therefore, a matrix that is multiplied by another matrix will give . This means that when a 2 by 3 matrix is multiplied by a 3 by 1 matrix, the product is a 2 by 1 matrix.

Matrix multiplication calculation

Since we now know that matrices are multiplied only when the number of columns of the first matrix equals the number of rows of the second matrix, then how do we calculate the product? It is actually simple. We shall follow some steps:

Step 1: This is done by matching each row by the corresponding column.

the letters in bold are those that are matched.

Step 2: Once matching is done, multiply each corresponding element. Again, this is based on position. So, the first element in the column of the first matrix multiplies the first element in the corresponding row of the second matrix. You do the same for the second element, the third element, and so on.

Step 3: After multiplying, then add up your products.

Step 4: Repeat the same process for other positions.

You should use this method no matter how large the column or row is.

Find the product of matrix and matrix

Solution:

Matrix T can be written as T2×3 and matrix U can be written as U3×2. Now, is T2×3 and U3×2 commutable? Yes, they both are. The new matrix should be TU2×2. So our product is a 2 by 2 matrix.

Now let us solve and confirm this;

TU is indeed a 2 by 2 matrix.

Coding in multiplying matrices

The coding system is another way of multiplying matrices. It emphasizes more on the position of the elements in the matrix. To apply the coding system you must know the dimension of the product matrix. It also helps you easily find the element at any position of the product matrix without necessarily finding elements in all positions of the product matrix.

If you have a matrixand matrix, then we expect their product SP to be a 2 by 2 matrix. We shall represent each position by the row and column it signifies.

Therefore;

Where;

r1 is row 1 or the first row

r2 is row 2 or the second row

c1 is column 1 or the first column

and

c2 is column 2 or the second column

r1c1 means the sum of the product of the first row of matrix S and the first column of matrix P

r1c2 means the sum of the product of the first row of matrix S and the second column of the matrix.

The same meaning goes for the rest.

The coding system helps guide you to prevent common linear coordinating errors when multiplying matrices. It also helps to find elements of a specific position in the product matrix.

If the product between matrix and matrix gives matrix, find the values of x and y.

Solution:

Using the coding system the product is

To find x, we know that its code in the matrix is r1c2. Recall that r1c2 is row 1 of matrix A multiplied by column 2 of matrix C. Thus

Also, to find y, we know that its code in the matrix is r2c3. Recall that r2c3 is row 2 of matrix A multiplied by column 3 of matrix C. Thus

Transposition of a matrix

To transpose a matrix, each column of the matrix is swapped to replace its corresponding row. A T is added at the top right corner of the matrix to show that the matrix is being transposed e.g. the transpose of a matrix A is expressed as AT.

If matrix, find BT.

Solution:

BT is the transpose of matrix B.

Note the bold numbers to keep track of how column 1 of matrix B became row 1 of matrix BT.

Solving word problems with matrices

Sometimes, word problems can be solved with matrices. The example below explains this.

Ireti sold 50 bags of maize, 20 bags of wheat and 30 bags of millet during the World Farmer's Fair. If a bag of maize, wheat and millet costs £100, £50 and £200 respectively, how much did Ireti make?

Solution:

Step 1: Represent the amount of bags for each food product in a 1 by 3 matrix.

Step 2: Represent the price per bag of each crop in a 3 by 1 matrix.

Step 3: Since you wish to find the total amount she made, it means the sum of the product of each crop by its respective price. This is represented as the product of the 1 by 3 matrix and the 3 by 1 matrix.

This means that Ireti made a total of £12,000 in her sales during the World Farmer's Fair.

Examples on operations with matrices

Generally, understanding is improved with more practice. Here are more examples to further elaborate on what you have learned so far.

Solve for

if

Solution:

Since 4P was given an we need to find 2P, we just need to divide the matrix 4P by 2 to obtain 2P. Hence,

Note that -R has already been, so you should just add 2P to -R.

No need to reconvert to R since the minus sigh goes through all elements of the matrix.

Hence,

Solve the following:

Solution:

The first matrix is a 3 by 3 matrix while the second is a 2 by 3 matrix. Basic operations such as addition and subtraction cannot be carried out when two matrices are not of the same dimension. Hence, both matrices cannot be combined to form a single matrix.

I hope you hadn't started adding any elements?

Operations with Matrices - Key takeaways

  • A matrix is just a rectangular or square collection of numbers that have been arranged in rows and columns.
  • An element is the individual number or variable which is arranged in a matrix.
  • Addition or subtraction of a matrix only occurs when matrices have the same dimension.
  • Multiplication of matrices only takes place when the number of columns of the first matrix equals the number of rows in the second matrix.
  • A coding system helps in the easy calculation of an element in a specific position of a product matrix.

Frequently Asked Questions about Operations with Matrices

These are the calculations carried out in a matrix system such as addition, subtraction, multiplication, and transposition.

Mixed operations or combined matrix operations are problems which require the application of more than one basic matrix operation. 

All matrix operations are done in arrays since you must consider both rows and columns of the matrix.

The matrix operation formulas are formulas used in carrying out operations in matrix. They vary depending on which operation is being carried out.

Final Operations with Matrices Quiz

Question

What condition must be met before adding or subtracting matrices?

Show answer

Answer

Matrices must be of the same dimensions.

Show question

Question

What condition must be met before matrices are multiplied?

Show answer

Answer

The number of columns of the first matrix must be equal to the number of rows of the second matrix.

Show question

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