How to Multiply the Same Matrix Over and Over Again

Matrix multiplication is one of the fundamental yet advanced concepts of matrices. You lot take to be careful while y'all multiply matrices. It is not as easy as it sounds. You lot should have sound knowledge of all the basic concepts like what a matrix is, the rows and columns in a matrix, how to represent a matrix, and how to multiply matrices.

Back to school: Matrices are the groupings of numbers, variables, symbols, or expressions in a rectangular table with varied numbers of rows and columns. They are rectangular arrays with different operations similar improver, multiplication, and transposition. The elements of a matrix are the numbers or entries that make up the matrix. The horizontal elements of matrices are rows, whereas the vertical elements are columns.

You lot can depict the number of rows and columns with the aid of variables. For example, let the number of rows in matrix K be 'chiliad' and the number of columns exist 'due north'. Thus we can represent the matrix Thou as [K] _{thousand x due north}. Matrix is represented within the square brackets '[ ]' with the product of its rows and columns in the subscript.

Until 1812 matrix multiplication was non known to mankind. Philippe Marie Binet, a French mathematician, invented matrix multiplication in 1812 to describe linear maps with matrices. He discovered that matrix multiplication is a binary operation. Ii matrices multiply to class a unmarried matrix.

Further in this article, yous shall learn the rules and major concepts for multiplying matrices.

What is Matrix Multiplication?

In the introduction of this article, you lot read that matrix multiplication is a binary functioning. This means that whenever you find the product of 2 or more than matrices, the answer is a single matrix, which follows the initial matrices. For multiplying two matrices, their compatibility is checked. This means that two matrices must follow a fix of rules to be multiplied.

These compatibility parameters are described below:

• The value of the number of columns in matrix one should be equal to the value of the number of rows of matrix 2.
• If the in a higher place condition doesn't satisfy, the matrices cannot be multiplied.
• If the matrices are square matrices with the same order, they can be multiplied by each other.
•  A square matrix of ii x two cannot be multiplied with a square matrix of iii x iii. It tin can be multiplied with another square matrix of 2 ten 2.

Permit u.s.a. have an example to understand this in a better fashion:

Instance one: Suppose nosotros take two matrices, K and 50. The order of the matrices is given as m x n and n x o, respectively. Find whether the matrices can be multiplied or non?

Solution: Nosotros are given

Matrix i = [ K ] _{m x northward}

Matrix 2 = [ L ] _{n x o}

We tin see that the number of columns in matrix One thousand equals the number of rows in matrix L. Hence the 2 matrices tin can be multiplied.

Resultant Matrix: The matrix achieved by multiplying two matrices is the resultant matrix. This matrix tin can be represented as:

• The proper noun of the matrix in square brackets
• The product of the number of rows of matrix 1 with the number of columns of matrix 2 as the order.

Again let us await at the example mentioned above:

We know that 1000 and L are compatible with multiplication. And so the resultant matrix, say D, would be = [ D ] _{g ten o} where m = number of rows of matrix one and o = the number of columns of matrix 2.

Example 2: A matrix [ M ] _{2 x 4} is multiplied by the matrix [ X ] _{4 ten 2}. If [ Y ] is the resultant matrix, then what is the order of Y.

Solution: Nosotros are given

Matrix 1 = [ Grand ] _{2 x four}

Matrix 2 = [ X ] _{4 ten 2}

Since the number of columns of matrix K equals the number of rows of matrix X, K and X tin can be multiplied. Therefore,

Matrix Y = [ Y ] _{2 x ii}

The order of matrix Y is 2 ten 2. Y is a foursquare matrix.

The orientation of two matrices is likewise a crucial factor for determining the product of the matrices. If there are ii matrices, Thousand and L, then to find the production of K and L, yous must get-go write the matrix M and so write matrix 50. If you write the matrix L before matrix K, you will get a completely different matrix. Commutative law doesn't apply during matrix multiplication. This means KL ≠ LK.

Now that we are clear with the rules for the multiplication of 2 matrices, let u.s. learn how to multiply matrices.

How to Multiply Matrices?

Let us larn the stepwise procedure for multiplying two matrices. At first, you may find it confusing simply when you get the hang of it, multiplying matrices is as easy as applying butter to your toast.

Footstep one: Check the compatibility of the matrices given. If they are not compatible, leave the multiplication.

Step 2: Take the first row of matrix 1 and multiply it with the get-go column of matrix two. Then multiply the first row of matrix 1 with the 2d column of matrix two. Now multiply the first row of matrix ane with the 3rd cavalcade of matrix 2, and so on. The values obtained from these volition fill the first row of the production matrix.

Footstep 3: At present take the second row of matrix one and multiply information technology with the first column of matrix 2 and follow the same steps as 2. This will fill the 2nd row of the product matrix.

Footstep 4: Continue these steps by taking each row until the production matrix is obtained.

By multiplying rows and columns we mean that the elements present in those rows and columns will exist multiplied. Remember to multiply the corresponding elements and and so add the products to detect the chemical element of the product matrix.

If you aren't sure virtually the steps then look at the pace by stride solved instance below:

Case: Find the product of the matrices given below:

K = and 50 =

Solution:

Step 1: The matrix G is of the order 1 x 4 and the matrix L is of the order 4 x two. Since the number of columns of matrix K matches with the number of rows of matrix 50, therefore the matrices are uniform for multiplication. The resultant matrix is of the club 1 x 2.

Step 2: Since K has only ane row, multiply it with the first column of matrix L in this manner: (ii 10 4) + (4 x 5) + (1 x 4) + (7 x three) = 51. Note how the 1st element of the row is multiplied with the 1st chemical element of column 1, similarly the second, 3rd, and 4th. They are and then added together to get the resultant matrix element.

Stride three: Too now multiply the 1st row with the 2d column of matrix Fifty. The outcome will exist (2 ten 3) + (4 x 2) + (1 x nine) + (seven x six) = 65.

Step 4: Arrange the issue in the product matrix. Let's suppose [ X ] is the resultant matrix; therefore, 10 = [51   65].

We just learned how to multiply ii matrices. The next section volition acquire how to solve a ( 2 x ii ) square matrix.

We just learned how to multiply two matrices. The side by side department will larn how to solve a ( 2 10 two ) square matrix.

How to Multiply 2 x 2 Matrix

For multiplying matrices ii x 2, you should exist well versed with the steps mentioned in the above section. Since we are multiplying 2 square matrices of the same order, we don't need to cheque the compatibility in this example. Remember that the product matrix will also be in the same society as the square matrix. Allow u.s.a. solve an example to sympathise how to multiply the matrix of 2 10 2.

Example: Multiply the square matrix

A = and B =

Step 1: Multiplying the first row of matrix A with the first column of matrix B. We go (two 10 1) + (9 x 3) = 29. Now multiplying the offset row of matrix A with the second column of matrix B. We become (2 x -4) + (ix x 7) = -8 + 63 = 55.

Stride 2: We will repeat step 1 merely change the row of matrix A. The kickoff and second columns of matrix B will exist multiplied by row 2 of matrix A. The results are

(iii x ane) + (-seven ten 3) = 3 – 21 = -eighteen, and (3 ten -4) + (-7 x 7) = -12 – 49 = -61

Step iii: Place the results properly

.

Therefore the resultant matrix, C =

Similarly, you lot can solve foursquare matrices like 3 x 3, iv x four, then on.

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