THE mmatrix multiplication is done through an algorithm that requires a lot of attention. For the product between matrix A and matrix B to exist, it is necessary that the number of columns gives first headquarters, in case A, is equal to the number of lines gives Monday headquarters, in case B.
From the multiplication between matrices, it is possible to understand what the identity matrix is, which is the neutral element of matrix multiplication, and what is the inverse matrix of the matrix M, which is the matrix M-1 whose product of M by M-1 is equal to the identity matrix. It is also possible to multiply a matrix by a real number — in this case, we multiply each of the terms of the headquarters by number.
Read too: What is a triangular matrix?
condition of existence
To multiply two matrices, first it is necessary to check the existence condition. For the product to exist, the number of columns in the first matrix must equal the number of rows in the second matrix.
Furthermore, the result of the multiplication is a matrix that has the same number of rows as the first matrix and the same number of columns as the second matrix.For example, the product AB between matrices A3x2 and B2x5 exists because the number of columns in A (2 columns) is equal to the number of rows in B (2 rows), and the result is matrix AB3x5. Already product between C matrices3x5 and matrix D2x5 does not exist, as C has 5 columns and D has 3 rows.
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How to calculate the product between two matrices?
To perform matrix multiplication, it is necessary to follow some steps. We will make an example of the multiplication of an algebraic matrix A2x3 by matrix B3x2
We know the product exists, because matrix A has 3 columns, and matrix B, 3 rows. We will call C the result of the multiplication A·B. In addition, we also know that the result is a C matrix.2x2, because matrix A has 2 rows, and matrix B, 2 columns.
To calculate the product of matrix A2x3 and matrix B3x2, let's follow a few steps.
First we will find each of the terms of the matrix C2x2:
To find the terms, let's always relate the rows of matrix A to the columns of matrix B:
ç11 → 1st line of A and 1st column of B
ç12 → 1st line of A and 2nd column of B
ç21 → 2nd line of A and 1st column of B
ç22 → 2nd line of A and 2nd column of B
We calculate each of the terms by multiplying the terms in the row of A and the terms in the column of B. Now we must add these products, starting with ç11:
1st line of A
1st column of B
ç11 = The11·B11 + The12·B21+ The13·B31
calculating ç12:
1st line of A
2nd column of B
ç12 = The11·B12 + The12·B22+The13·B32
calculating ç21:
2nd line of A
1st column of B
ç21 = The21·B11 + The22·B21+The23·B31
calculating the term ç22:
2nd line of A
2nd column of B
ç22 = The21·B12 + The22·B22+The23·B32
Thus, matrix C is formed by the terms:
Example:
Let's calculate the multiplication between matrices A and B.
We know that in A2x2 and B2x3, the number of columns in the first is equal to the number of rows in the second, so the product exists. So we will make C = A· B and we know that C2x3.
Multiplying, we have to:
See too: What is a transposed matrix?
identity matrix
In multiplication between matrices, there are some special cases, such as the identity matrix, which is the neutral element of multiplication between matrices.. The identity matrix is a square matrix, that is, the number of rows is always equal to the number of columns. Furthermore, only the terms of the diagonal are equal to 1 in it, and the other terms are all equal to zero. When we multiply a matrix M by the identity matrix Ino, We have to:
M · Ino = M
Example:
What is the inverse matrix?
Given a matrix M, we know it as an inverse matrix of M. the matrix M-1whose product M · M-1 equals à identity matrix Ino. For a matrix to have an inverse, it must be square, and its determinant has to be different from 0. Let's look at examples of matrices that are inverse:
Calculating the product A·B, we have to:
Note that the product between A and B generated matrix I2. When this happens, we say that B is the inverse matrix of A. To learn more about this type of matrix, read: Inverse matrix.
Matrix multiplication by a real number
Unlike multiplication between matrices, there is also matrix multiplication by one real number, which is a much simpler operation to find the solution.
Given a matrix M, multiplying the matrix by a real number k is equal to the matrix kM. To find this matrix kM, enough multiply all terms in the matrix by the constant k.
Example:
if k = 5 and considering matrix M below, find matrix 5M.
Multiplying:
solved exercises
Question 1 - (Unitau) Given matrices A and B,
the value of element c11 of matrix C = AB is:
A) 10.
B) 28.
C) 38.
D) 18.
E) 8.
Resolution
Alternative A.
How do we want the term c11, let's multiply the terms in the first row and A with the terms in the first column of B.
calculating c11 = 1 · 3 + 2 · 2 + 3 · 1 = 3 + 4 + 3 = 10
Question 2 - (Enem 2012) A student registered the bimonthly grades of some of his subjects in a table. He noted that the numerical entries in the table formed a 4 × 4 matrix, and that he could calculate the annual averages for these disciplines using the product of matrices. All the tests had the same weight, and the table he got is shown below.
To obtain these averages, he multiplied the matrix obtained from the table by the matrix:
Resolution
Alternative E.
The average is nothing more than the sum of elements divided by the number of elements. Note that there are 4 notes per line, so the average would be the sum of those notes divided by 4. Dividing by 4 is the same as multiplying by fraction ¼. Also, the matrix of grades is a 4x4 matrix, so we have to multiply by a 4x1 matrix, that is, it has 4 rows and 1 column, to find the matrix that has the average of the grades.
By Raul Rodrigues de Oliveira
Maths teacher
