Select The Correct Answer.What Is The Result Of The Operation $-2\left[\begin{array}{ccc}-3 & 5 & 1 \ 8 & -2 & 3 \ 2 & 1 & -4\end{array}\right]?A. $\left[\begin{array}{ccc}6 & -10 & -2 \ -16 & 4 & 6 \ -4 & -2 &

Introduction

In mathematics, matrices are a fundamental concept in linear algebra, and understanding how to perform operations on them is crucial for solving various problems. One of the basic operations on matrices is multiplying a matrix by a scalar. In this article, we will explore the result of multiplying a given matrix by a scalar, specifically the operation $-2\left[\begin{array}{ccc}-3 & 5 & 1 \\ 8 & -2 & 3 \\ 2 & 1 & -4\end{array}\right]$.

What is a Matrix?

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. It is a powerful tool for representing systems of equations, transformations, and other mathematical concepts. Matrices can be used to solve systems of linear equations, find the inverse of a matrix, and perform various other operations.

Multiplying a Matrix by a Scalar

When we multiply a matrix by a scalar, we are essentially multiplying each element of the matrix by that scalar. This operation is denoted by the symbol $\times$ or $\cdot$. For example, if we have a matrix $A = \left[\begin{array}{ccc}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$ and we want to multiply it by a scalar $k$, the resulting matrix will be $kA = \left[\begin{array}{ccc}ka & kb & kc \\ kd & ke & kf \\ kg & kh & ki\end{array}\right]$.

**The Operation $-2\left[\begin{array}{ccc}-3 & 5 & 1 \\ 8 & -2 & 3 \\ 2 & 1 & -4\end{array}\right]$

Now, let's apply this concept to the given operation $-2\left[\begin{array}{ccc}-3 & 5 & 1 \\ 8 & -2 & 3 \\ 2 & 1 & -4\end{array}\right]$. To find the result, we need to multiply each element of the matrix by the scalar $-2$.

**Step 1: Multiply the first row by $-2$

The first row of the matrix is $[-3, 5, 1]$. Multiplying each element by $-2$, we get $[-6, -10, -2]$.

**Step 2: Multiply the second row by $-2$

The second row of the matrix is $[8, -2, 3]$. Multiplying each element by $-2$, we get $[-16, 4, 6]$.

**Step 3: Multiply the third row by $-2$

The third row of the matrix is $[2, 1, -4]$. Multiplying each element by $-2$, we get $[-4, -2, 8]$.

The Resulting Matrix

After multiplying each row by $-2$, the resulting matrix is:

$$\left[\begin{array}{ccc}-6 & -10 & -2 \\ -16 & 4 & 6 \\ -4 & -2 & 8\end{array}\right]$$

Conclusion

In conclusion, the result of the operation $-2\left[\begin{array}{ccc}-3 & 5 & 1 \\ 8 & -2 & 3 \\ 2 & 1 & -4\end{array}\right]$ is the matrix $\left[\begin{array}{ccc}-6 & -10 & -2 \\ -16 & 4 & 6 \\ -4 & -2 & 8\end{array}\right]$. This demonstrates the concept of multiplying a matrix by a scalar, which is a fundamental operation in linear algebra.

Example Use Cases

Multiplying a matrix by a scalar has various applications in mathematics and other fields. Some examples include:

• Scaling transformations: Multiplying a matrix by a scalar can be used to scale a transformation, such as scaling a 2D or 3D object.
• Linear equations: Multiplying a matrix by a scalar can be used to solve systems of linear equations.
• Data analysis: Multiplying a matrix by a scalar can be used to normalize or scale data in data analysis.

Tips and Tricks

When working with matrices, it's essential to remember the following tips and tricks:

• Pay attention to the scalar: When multiplying a matrix by a scalar, make sure to multiply each element by the scalar.
• Use the correct operation: When multiplying a matrix by a scalar, use the correct operation, such as $\times$ or $\cdot$.
• Check your work: When multiplying a matrix by a scalar, double-check your work to ensure that each element is multiplied correctly.

Introduction

In our previous article, we explored the concept of multiplying a matrix by a scalar. In this article, we will provide a Q&A guide to help you better understand matrix operations and how to apply them in various situations.

Q: What is a matrix?

A: A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. It is a powerful tool for representing systems of equations, transformations, and other mathematical concepts.

Q: What is the difference between a matrix and a vector?

A: A vector is a one-dimensional array of numbers, while a matrix is a two-dimensional array of numbers. Vectors are often used to represent points in space, while matrices are used to represent systems of equations or transformations.

Q: How do I multiply a matrix by a scalar?

A: To multiply a matrix by a scalar, you need to multiply each element of the matrix by the scalar. This operation is denoted by the symbol $\times$ or $\cdot$. For example, if you have a matrix $A = \left[\begin{array}{ccc}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$ and you want to multiply it by a scalar $k$, the resulting matrix will be $kA = \left[\begin{array}{ccc}ka & kb & kc \\ kd & ke & kf \\ kg & kh & ki\end{array}\right]$.

Q: What is the result of multiplying a matrix by a scalar?

A: The result of multiplying a matrix by a scalar is a new matrix where each element is multiplied by the scalar. The resulting matrix has the same dimensions as the original matrix.

Q: Can I multiply a matrix by a vector?

A: Yes, you can multiply a matrix by a vector. This operation is called the matrix-vector product. The result of this operation is a new vector where each element is the dot product of the corresponding row of the matrix and the vector.

Q: What is the difference between the matrix-vector product and the matrix-matrix product?

A: The matrix-vector product is the result of multiplying a matrix by a vector, while the matrix-matrix product is the result of multiplying two matrices. The matrix-matrix product is a more complex operation that involves multiplying each element of the first matrix by the corresponding element of the second matrix.

Q: How do I perform the matrix-matrix product?

A: To perform the matrix-matrix product, you need to multiply each element of the first matrix by the corresponding element of the second matrix. This operation is denoted by the symbol $\times$ or $\cdot$. For example, if you have two matrices $A = \left[\begin{array}{ccc}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$ and $B = \left[\begin{array}{ccc}j & k & l \\ m & n & o \\ p & q & r\end{array}\right]$, the resulting matrix will be $AB = \left[\begin{array}{ccc}aj+bm+cp & ak+bn+cq & al+bo+cr \\ dj+em+fp & dk+en+fq & dl+eo+fr \\ gj+hm+ip & gk+hn+iq & gl+ho+ir\end{array}\right]$.

Q: What is the result of the matrix-matrix product?

A: The result of the matrix-matrix product is a new matrix where each element is the sum of the products of the corresponding elements of the two input matrices.

Q: Can I perform the matrix-matrix product with matrices of different dimensions?

A: No, you cannot perform the matrix-matrix product with matrices of different dimensions. The number of columns in the first matrix must be equal to the number of rows in the second matrix.

Q: What are some common applications of matrix operations?

A: Matrix operations have many applications in various fields, including:

• Linear algebra: Matrix operations are used to solve systems of linear equations, find the inverse of a matrix, and perform various other operations.
• Data analysis: Matrix operations are used to normalize or scale data, perform principal component analysis, and other data analysis techniques.
• Computer graphics: Matrix operations are used to perform transformations, such as scaling, rotating, and translating objects.
• Machine learning: Matrix operations are used to perform various machine learning algorithms, such as neural networks and support vector machines.

Conclusion

In conclusion, matrix operations are a fundamental concept in mathematics and have many applications in various fields. By understanding how to perform matrix operations, you can solve various problems and apply them in real-world situations.