Matrix Size and Multiplication

Oct 13, 2025 by talynt.org 104 views

=====================================================

Introduction

Matrix multiplication is a fundamental concept in linear algebra, and it plays a crucial role in various fields such as physics, engineering, and computer science. In this article, we will explore the concept of matrix multiplication, and we will use the given matrices $X$ and $Y$ to demonstrate the process.

What is Matrix Multiplication?

Matrix multiplication is a way of combining two matrices to produce a new matrix. The resulting matrix is obtained by multiplying the elements of the rows of the first matrix with the elements of the columns of the second matrix. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.

Matrix Size and Multiplication

Before we can multiply two matrices, we need to check if the matrices can be multiplied. The number of columns in the first matrix must be equal to the number of rows in the second matrix. If this condition is met, then the matrices can be multiplied.

Given Matrices

Let's consider the given matrices $X$ and $Y$ :

$X=\left[\begin{array}{ll}-4 & -3\end{array}\right]$

$Y=\left[\begin{array}{rr}-4 & 1 \\ -2 & 7\end{array}\right]$

Checking Matrix Size

To check if the matrices can be multiplied, we need to compare the number of columns in the first matrix with the number of rows in the second matrix. The first matrix $X$ has 2 columns, and the second matrix $Y$ has 2 rows. Since the number of columns in the first matrix is equal to the number of rows in the second matrix, the matrices can be multiplied.

Matrix Multiplication

Now that we have checked the matrix size, we can proceed with the matrix multiplication. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. In this case, the resulting matrix will have 1 row and 2 columns.

To calculate the elements of the resulting matrix, we need to multiply the elements of the rows of the first matrix with the elements of the columns of the second matrix.

Calculating the Elements of the Resulting Matrix

Let's calculate the elements of the resulting matrix:

The element in the first row and first column is calculated as: $(-4)(-4) + (-3)(-2) = 16 + 6 = 22$
The element in the first row and second column is calculated as: $(-4)(1) + (-3)(7) = -4 - 21 = -25$

The Resulting Matrix

The resulting matrix is:

$XY = \left[\begin{array}{rr}22 & -25\end{array}\right]$

Conclusion

In this article, we have demonstrated the process of matrix multiplication using the given matrices $X$ and $Y$ . We have checked the matrix size and confirmed that the matrices can be multiplied. We have then calculated the elements of the resulting matrix and obtained the final result.

Matrix multiplication is a powerful tool in linear algebra, and it has numerous applications in various fields. By understanding the concept of matrix multiplication, we can solve complex problems and make informed decisions.

Final Answer

The final answer is $\boxed{\left[\begin{array}{rr}22 & -25\end{array}\right]}$ .

=====================================================

Introduction

What is Matrix Multiplication?

Matrix Size and Multiplication

Given Matrices

Let's consider the given matrices $X$ and $Y$ :

$X=\left[\begin{array}{ll}-4 & -3\end{array}\right]$

$Y=\left[\begin{array}{rr}-4 & 1 \\ -2 & 7\end{array}\right]$

Checking Matrix Size

Matrix Multiplication

To calculate the elements of the resulting matrix, we need to multiply the elements of the rows of the first matrix with the elements of the columns of the second matrix.

Calculating the Elements of the Resulting Matrix

Let's calculate the elements of the resulting matrix:

The element in the first row and first column is calculated as: $(-4)(-4) + (-3)(-2) = 16 + 6 = 22$
The element in the first row and second column is calculated as: $(-4)(1) + (-3)(7) = -4 - 21 = -25$

The Resulting Matrix

The resulting matrix is:

$XY = \left[\begin{array}{rr}22 & -25\end{array}\right]$

Q&A

Q: What is matrix multiplication?

A: Matrix multiplication is a way of combining two matrices to produce a new matrix. The resulting matrix is obtained by multiplying the elements of the rows of the first matrix with the elements of the columns of the second matrix.

Q: What are the conditions for matrix multiplication?

A: The number of columns in the first matrix must be equal to the number of rows in the second matrix. If this condition is met, then the matrices can be multiplied.

Q: How do we calculate the elements of the resulting matrix?

A: To calculate the elements of the resulting matrix, we need to multiply the elements of the rows of the first matrix with the elements of the columns of the second matrix.

Q: What is the size of the resulting matrix?

A: The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.

Q: Can we multiply two matrices if they have different sizes?

A: No, we cannot multiply two matrices if they have different sizes. The number of columns in the first matrix must be equal to the number of rows in the second matrix.

Q: What is the resulting matrix when we multiply two matrices?

A: The resulting matrix is obtained by multiplying the elements of the rows of the first matrix with the elements of the columns of the second matrix.

Conclusion

Final Answer

The final answer is $\boxed{\left[\begin{array}{rr}22 & -25\end{array}\right]}$ .