Problem 47 - Algebra 2

Question

Perform the indicated matrix operations. If the matrix does not exist, write impossible. 3\(\left[\begin{array}{rr}{4} & {-2} \\ {-1} & {7}\end{array}\right]\)

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} 12 & -6 \\ -3 & 21 \end{bmatrix} \).

Step 1: Understand the Matrix and Scalar Multiplication

We are given a 2x2 matrix \( \begin{bmatrix} 4 & -2 \ -1 & 7 \end{bmatrix} \) and a scalar 3. The task is to perform scalar multiplication, which involves multiplying each element of the matrix by 3.

Step 2: Multiply Each Element by the Scalar

Multiply each element of the matrix by the scalar 3: \[3 \times \begin{bmatrix} 4 & -2 \ -1 & 7 \end{bmatrix} = \begin{bmatrix} 3 \times 4 & 3 \times -2 \ 3 \times -1 & 3 \times 7 \end{bmatrix}\]Simplifying, we get:\[\begin{bmatrix} 12 & -6 \ -3 & 21 \end{bmatrix}\]

Step 3: Present the Result

After performing the scalar multiplication for each element, the resulting matrix is:\[ \begin{bmatrix} 12 & -6 \ -3 & 21 \end{bmatrix} \]Thus, the operation is possible and results in a 2x2 matrix with these elements.

Key Concepts

Scalar Multiplication

Scalar multiplication is a basic matrix operation. It involves multiplying each element of a matrix by a constant, known as a scalar. In this operation, we keep the structure of the matrix unchanged, but modify the elements within it.
For instance, if we are given a scalar (say, 3) and a matrix \[\begin{bmatrix} 4 & -2 \ -1 & 7 \end{bmatrix}\], we multiply each matrix element by 3. This results in a new matrix with elements being the product of the original values and the scalar. You do not change the matrix size, just the values inside.

This operation is useful in a variety of contexts, allowing us to adjust the scale, or magnitude, of a matrix without altering its fundamental structure. Whether adapting data or performing calculations in physics or economics, scalar multiplication is a key tool. Remember, the operation is straightforward and lacks complexity: it's all about consistent scaling across all elements.

2x2 Matrix

A 2x2 matrix is a specific type of matrix with exactly two rows and two columns. It's often the simplest example when learning matrix operations. The matrix looks like this:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]
Here, "a", "b", "c", and "d" are the elements of the matrix.

These matrices are incredibly versatile in mathematical computations. They are particularly common when dealing with transformations in geometry, like rotations and reflections in a two-dimensional space.

One of the key features of a 2x2 matrix is its size ease. They're manageable for manual computations and a good stepping stone towards understanding larger matrices.

Matrix Operations

Matrix operations include several calculations involving matrices, such as addition, subtraction, and multiplication (both scalar and matrix multiplication).

Each operation follows specific rules. For scalar multiplication, as discussed earlier, we multiply every element of the matrix by the scalar. If you are given another matrix of the same dimension, you can also add or subtract corresponding elements for addition or subtraction operations.

Matrix multiplication, which is different from scalar multiplication, involves a dot product between rows and columns. However, it's essential to note that not all matrices can be multiplied in this way unless their dimensions align correctly.

Matrix operations are foundational in linear algebra and have applications spanning physics, engineering, computer science, and more. They enable the description and resolution of complex systems and transformations, and learning them paves the way for advanced mathematical proficiency.