Given The Matrices $A$ And $B$ Shown Below, Find $-4A - 6B$.$\[ A = \begin{bmatrix} -4 & 6 & -4 \end{bmatrix} \quad B = \begin{bmatrix} -1 & -1 & 2 \end{bmatrix} \\](Note: The Problem Refers To Operations On 1x3

Matrix Operations: Finding $-4A - 6B$

In linear algebra, matrices are used to represent systems of equations and perform various operations. Given two matrices $A$ and $B$, we can perform scalar multiplication and addition/subtraction to obtain new matrices. In this article, we will explore how to find the matrix $-4A - 6B$ using the given matrices $A$ and $B$.

The given matrices are:

${ A = \begin{bmatrix} -4 & 6 & -4 \end{bmatrix} }$

${ B = \begin{bmatrix} -1 & -1 & 2 \end{bmatrix} }$

To find $-4A$ and $-6B$, we need to perform scalar multiplication. Scalar multiplication involves multiplying each element of a matrix by a scalar.

${ -4A = -4 \begin{bmatrix} -4 & 6 & -4 \end{bmatrix} = \begin{bmatrix} 16 & -24 & 16 \end{bmatrix} }$

${ -6B = -6 \begin{bmatrix} -1 & -1 & 2 \end{bmatrix} = \begin{bmatrix} 6 & 6 & -12 \end{bmatrix} }$

Now that we have $-4A$ and $-6B$, we can find $-4A - 6B$ by adding/subtracting the corresponding elements of the two matrices.

${ -4A - 6B = \begin{bmatrix} 16 & -24 & 16 \end{bmatrix} - \begin{bmatrix} 6 & 6 & -12 \end{bmatrix} }$

${ = \begin{bmatrix} 16 - 6 & -24 - 6 & 16 - (-12) \end{bmatrix} }$

${ = \begin{bmatrix} 10 & -30 & 28 \end{bmatrix} }$

In this article, we found the matrix $-4A - 6B$ using the given matrices $A$ and $B$. We performed scalar multiplication to find $-4A$ and $-6B$, and then added/subtracted the corresponding elements to find the final matrix. This demonstrates the importance of understanding matrix operations in linear algebra.

• Scalar multiplication involves multiplying each element of a matrix by a scalar.
• Matrix addition/subtraction involves adding/subtracting the corresponding elements of two matrices.
• Matrix operations can be used to solve systems of equations and perform various calculations.

Matrix operations have numerous real-world applications, including:

• Computer Graphics: Matrix operations are used to perform transformations on 2D and 3D objects.
• Machine Learning: Matrix operations are used to perform linear algebra operations on large datasets.
• Data Analysis: Matrix operations are used to perform statistical analysis and data visualization.

In our previous article, we explored how to find the matrix $-4A - 6B$ using the given matrices $A$ and $B$. In this article, we will answer some frequently asked questions about matrix operations.

Q: What is scalar multiplication?

A: Scalar multiplication involves multiplying each element of a matrix by a scalar. For example, if we have a matrix $A = \begin{bmatrix} 2 & 3 & 4 \end{bmatrix}$ and a scalar $k = 3$, then the scalar multiplication of $A$ and $k$ is given by:

${ kA = 3 \begin{bmatrix} 2 & 3 & 4 \end{bmatrix} = \begin{bmatrix} 6 & 9 & 12 \end{bmatrix} }$

Q: What is matrix addition/subtraction?

A: Matrix addition/subtraction involves adding/subtracting the corresponding elements of two matrices. For example, if we have two matrices $A = \begin{bmatrix} 2 & 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 & 7 \end{bmatrix}$, then the matrix addition of $A$ and $B$ is given by:

${ A + B = \begin{bmatrix} 2 & 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 & 7 \end{bmatrix} = \begin{bmatrix} 7 & 9 & 11 \end{bmatrix} }$

Q: Can I add/subtract matrices of different sizes?

A: No, you cannot add/subtract matrices of different sizes. The matrices must have the same number of rows and columns in order to perform addition/subtraction.

Q: Can I multiply matrices of different sizes?

A: No, you cannot multiply matrices of 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 difference between matrix multiplication and scalar multiplication?

A: Matrix multiplication involves multiplying the elements of two matrices, while scalar multiplication involves multiplying each element of a matrix by a scalar.

Q: Can I use matrix operations to solve systems of equations?

A: Yes, matrix operations can be used to solve systems of equations. For example, if we have a system of equations:

${ 2x + 3y + 4z = 10 }$ ${ 5x + 6y + 7z = 20 }$

We can represent this system as a matrix equation:

${ \begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 7 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 10 \\ 20 \end{bmatrix} }$

We can then use matrix operations to solve for $x$, $y$, and $z$.

In this article, we answered some frequently asked questions about matrix operations. We covered topics such as scalar multiplication, matrix addition/subtraction, and matrix multiplication. We also discussed the importance of matrix operations in solving systems of equations.