Problem 26 - Algebra 2

Question

Perform the indicated matrix operation. If the matrix does not exist, write impossible. $$ 3\left[\begin{array}{rr}{5} & {-3} \\ {-10} & {8} \\ {-1} & {7}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \(\begin{bmatrix} 15 & -9 \\ -30 & 24 \\ -3 & 21 \end{bmatrix}\).

Step 1: Understand the Problem

We are given a scalar multiplication problem where we need to multiply a 3x2 matrix by the scalar 3. The matrix is:\[\begin{bmatrix}5 & -3 \-10 & 8 \-1 & 7\end{bmatrix}\].

Step 2: Perform Scalar Multiplication

To perform scalar multiplication, multiply each element of the matrix by the scalar 3. Let's calculate each element:- First row: \(3 \times 5 = 15\), \(3 \times (-3) = -9\)- Second row: \(3 \times (-10) = -30\), \(3 \times 8 = 24\)- Third row: \(3 \times (-1) = -3\), \(3 \times 7 = 21\)

Step 3: Write the Resulting Matrix

After performing the scalar multiplication, the resulting matrix is:\[\begin{bmatrix}15 & -9 \-30 & 24 \-3 & 21\end{bmatrix}\].

Key Concepts

Matrix Operations

Matrix operations are a fundamental part of linear algebra and include various actions such as addition, subtraction, scalar multiplication, and more. In this exercise, the focus is on **scalar multiplication**. This operation involves multiplying every element of a matrix by a specific number, known as a scalar.

When performing scalar multiplication, the process is straightforward:

• Identify the scalar you need to multiply with. In this case, it is 3.
• Multiply the scalar with each entry of the matrix individually.
• Write down the new matrix with the resulting products.

Scalar multiplication maintains the dimensions of the original matrix. If you start with a 3x2 matrix, after multiplying each element by a scalar, you will still have a 3x2 matrix.

3x2 Matrix

A 3x2 matrix, although simple, has specific characteristics. It is a rectangular array consisting of three rows and two columns. Each entry or element of the matrix is defined by its row and column position.

For example, in the matrix:

• The element 5 is in the first row, first column.
• The element -3 is in the first row, second column.
• The element -10 is in the second row, first column, and so on.

When it comes to matrix operations like scalar multiplication, understanding and correctly identifying these rows and columns is crucial. This type of matrix is commonly used in Algebra 2 problems as it is simple enough to allow for easy computation, yet complex enough to illustrate the mechanics of matrix operations.

Algebra 2

Algebra 2 extends the concepts introduced in Algebra 1, including more advanced topics such as matrices. In Algebra 2, matrices are used to solve systems of equations, perform transformations, and much more. Scalar multiplication is one of the basic operations on matrices within this field.

This exercise is a practical example of how Algebra 2 uses matrices. By practicing operations like scalar multiplication, students can build a foundation for more complex topics, including matrix addition, matrix multiplication, and solving matrix equations.
Algebra 2 problems often require students to visualize and manipulate matrices, which aids in understanding not only theoretical concepts but also practical applications. Scalar multiplication is a stepping stone to grasping the broader use of matrices in real-life situations, such as computer graphics and economic modeling.