Product of Identity Matrix and Scalar

Oct 13, 2025 by windailys.com 104 views

Introduction

Matrix multiplication and scalar multiplication are fundamental concepts in linear algebra. In this article, we will explore the product of a $$3 \times 3$$ identity matrix and the scalar $$3$$. We will examine the properties of matrix multiplication and scalar multiplication to determine the correct answer.

Matrix Multiplication

Matrix multiplication is a binary operation that takes two matrices as input and produces another matrix as output. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix. The elements of the resulting matrix are calculated by multiplying the elements of the rows of the first matrix with the elements of the columns of the second matrix.

Identity Matrix

An identity matrix is a square matrix that has 1s on the main diagonal and 0s elsewhere. The identity matrix is denoted by $$I_n$$, where $$n$$ is the size of the matrix. The identity matrix has the following properties:

When multiplied by any matrix, the result is the same matrix.
When multiplied by the identity matrix, the result is the same matrix.

Scalar Multiplication

Scalar multiplication is a binary operation that takes a scalar and a matrix as input and produces another matrix as output. The resulting matrix has the same number of rows and columns as the original matrix. The elements of the resulting matrix are calculated by multiplying the elements of the original matrix by the scalar.

Product of Identity Matrix and Scalar

Now, let's consider the product of a $$3 \times 3$$ identity matrix and the scalar $$3$$. The identity matrix is:

\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]

When we multiply this matrix by the scalar $$3$$, we get:

3 \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] = \left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{array}\right]

Conclusion

In conclusion, the product of a $$3 \times 3$$ identity matrix and the scalar $$3$$ is:

\left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{array}\right]

This is the correct answer.

Answer

The correct answer is:

A. $$\left[\begin{array}{lll}3 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 3\end{array}\right]$$

Discussion

This problem requires a good understanding of matrix multiplication and scalar multiplication. The key concept is that the identity matrix does not change when multiplied by any matrix, and when multiplied by a scalar, the result is the same matrix multiplied by the scalar.

Example Use Cases

Matrix multiplication and scalar multiplication have many applications in various fields, including:

Linear algebra: Matrix multiplication is used to solve systems of linear equations, find the inverse of a matrix, and calculate the determinant of a matrix.
Computer graphics: Matrix multiplication is used to perform transformations on 2D and 3D objects, such as rotation, scaling, and translation.
Data analysis: Matrix multiplication is used to perform operations on large datasets, such as data compression and data encryption.

Conclusion

$In conclusion, matrix multiplication and scalar multiplication are fundamental concepts in linear algebra. Understanding these concepts is essential for solving problems involving matrices and vectors. The product of a $$3 \times 3$$ identity matrix and the scalar $$3$$ is a simple example of matrix multiplication and scalar multiplication.<br/> **Matrix Multiplication and Scalar Multiplication: Q&A**$

Introduction

Matrix multiplication and scalar multiplication are fundamental concepts in linear algebra. In this article, we will explore some common questions and answers related to matrix multiplication and scalar multiplication.

Q: What is matrix multiplication?

A: Matrix multiplication is a binary operation that takes two matrices as input and produces another matrix as output. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix. The elements of the resulting matrix are calculated by multiplying the elements of the rows of the first matrix with the elements of the columns of the second matrix.

Q: What is the identity matrix?

A: The identity matrix is a square matrix that has 1s on the main diagonal and 0s elsewhere. The identity matrix is denoted by $$I_n$$, where $$n$$ is the size of the matrix. The identity matrix has the following properties:

When multiplied by any matrix, the result is the same matrix.
When multiplied by the identity matrix, the result is the same matrix.

Q: What is scalar multiplication?

A: Scalar multiplication is a binary operation that takes a scalar and a matrix as input and produces another matrix as output. The resulting matrix has the same number of rows and columns as the original matrix. The elements of the resulting matrix are calculated by multiplying the elements of the original matrix by the scalar.

Q: How do I multiply two matrices?

A: To multiply two matrices, you need to follow these steps:

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.
Multiply the elements of the rows of the first matrix with the elements of the columns of the second matrix.
Add the products of the corresponding elements to get the elements of the resulting matrix.

Q: What is the product of a $$3 \times 3$$ identity matrix and the scalar $$3$$?

A: The product of a $$3 \times 3$$ identity matrix and the scalar $$3$$ is:

3 \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] = \left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{array}\right]

Q: What are some common applications of matrix multiplication and scalar multiplication?

A: Matrix multiplication and scalar multiplication have many applications in various fields, including:

Linear algebra: Matrix multiplication is used to solve systems of linear equations, find the inverse of a matrix, and calculate the determinant of a matrix.
Computer graphics: Matrix multiplication is used to perform transformations on 2D and 3D objects, such as rotation, scaling, and translation.
Data analysis: Matrix multiplication is used to perform operations on large datasets, such as data compression and data encryption.

Q: What are some common mistakes to avoid when working with matrix multiplication and scalar multiplication?

A: Some common mistakes to avoid when working with matrix multiplication and scalar multiplication include:

Not checking if the matrices can be multiplied.
Not following the correct order of operations when multiplying matrices.
Not using the correct scalar when multiplying a matrix by a scalar.

Conclusion

In conclusion, matrix multiplication and scalar multiplication are fundamental concepts in linear algebra. Understanding these concepts is essential for solving problems involving matrices and vectors. By following the steps outlined in this article, you can avoid common mistakes and become proficient in matrix multiplication and scalar multiplication.