Problem 78 - Algebra 2

Question

Perform the indicated operations. $$ -2\left[\begin{array}{ccc}{-3} & {0} & {12} \\ {-7} & {\frac{1}{3}} & {4}\end{array}\right] $$

Step-by-Step Solution

Answer

The result of the operation is \( \begin{bmatrix} 6 & 0 & -24 \\ 14 & -\frac{2}{3} & -8 \end{bmatrix} \).

Step 1: Understand Scalar Multiplication

To solve the given matrix operation, note that multiplying a scalar by a matrix means multiplying each element of the matrix by the scalar. In this case, we need to multiply each element of the matrix by \(-2\).

Step 2: Multiply Each Element by the Scalar

Take each element of the matrix and multiply it by \(-2\):- For the first row, multiply \(-3\) by \(-2\) to get \(6\).- Multiply \(0\) by \(-2\) to get \(0\).- Multiply \(12\) by \(-2\) to get \(-24\).- For the second row, multiply \(-7\) by \(-2\) to get \(14\).- Multiply \(\frac{1}{3}\) by \(-2\) to get \(-\frac{2}{3}\).- Multiply \(4\) by \(-2\) to get \(-8\).

Step 3: Write the Resultant Matrix

After performing the necessary operations in Step 2, the resultant matrix is:\[\begin{bmatrix}6 & 0 & -24 \14 & -\frac{2}{3} & -8\end{bmatrix}\]

Key Concepts

Matrix Operations

When dealing with matrices, it's essential to understand that they are just rectangular arrays of numbers organized into rows and columns. Matrix operations involve the various ways in which matrices can be combined or altered. There are several types of matrix operations: addition, subtraction, multiplication, scalar multiplication, and transposition, among others. Each operation follows specific rules.

• For addition and subtraction, matrices need to have the same dimensions. This means they must have the same number of rows and columns, and these operations are performed by adding or subtracting corresponding elements.
• Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. However, scalar multiplication, which is the focus here, is much simpler because it involves multiplying a matrix by a single number.

Matrix operations can be very different from basic arithmetic due to these rules, highlighting the specialized nature of dealing with matrices.

Scalar Multiplication

Scalar multiplication is one of the fundamental matrix operations. It involves multiplying each entry of a matrix by a constant number, known as the scalar. This operation is straightforward because every element within the matrix is treated the same way. The scalar affects every component equally, effectively "scaling" the entire matrix by that factor.Consider the matrix \[\begin{bmatrix}-3 & 0 & 12 \ -7 & \frac{1}{3} & 4\end{bmatrix}\] and the scalar ",