Problem 11 - Precalculus: Mathematics for Calculus

Question

Perform the matrix operation, or if it is impossible, explain why. $$3\left[\begin{array}{rr} 1 & 2 \\ 4 & -1 \\ 1 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The resulting matrix is \(\begin{bmatrix} 3 & 6 \\ 12 & -3 \\ 3 & 0 \end{bmatrix}\).

Step 1: Understand Scalar Multiplication

In this exercise, we are asked to multiply a scalar, which is \( 3 \), by a matrix. Scalar multiplication involves multiplying every element of the matrix by the scalar.

Step 2: Examine the Given Matrix

We are given the matrix \(\begin{bmatrix}1 & 2 \4 & -1 \1 & 0\end{bmatrix}\). It is a 3x2 matrix with three rows and two columns.

Step 3: Multiply Each Element by the Scalar

Multiply each element of the matrix by \( 3 \):\(3 \times \begin{bmatrix}1 & 2 \4 & -1 \1 & 0\end{bmatrix} = \begin{bmatrix}3 \times 1 & 3 \times 2 \3 \times 4 & 3 \times (-1) \3 \times 1 & 3 \times 0\end{bmatrix} = \begin{bmatrix}3 & 6 \12 & -3 \3 & 0\end{bmatrix}\).

Step 4: Finalize the Result

The final matrix after performing the scalar multiplication is \(\begin{bmatrix}3 & 6 \12 & -3 \3 & 0\end{bmatrix}\). There are no special rules or exceptions when multiplying a scalar by a matrix, so this operation is always possible.

Key Concepts

Matrix Operations

Matrix operations are a fundamental part of linear algebra and include various ways to interact with matrices. Common operations include addition, subtraction, and multiplication. Here, we focus on scalar multiplication, but it's useful to understand other operations as well:

• Addition: You can add two matrices if they have the same dimensions. Add corresponding elements from each matrix.
• Subtraction: Similarly, subtraction is done by subtracting corresponding elements of matrices of the same dimensions.
• Multiplication: Multiplying two matrices involves multiplying rows by columns, but that's different from scalar multiplication.

Matrix operations allow us to manipulate matrices to solve equations, transform data, and model various processes. Scalar multiplication is one of the simpler operations, performed without changing the structure, but scaling each element.

3x2 Matrix

A 3x2 matrix is a specific type of matrix that has three rows and two columns. When written in rectangular brackets, it looks like this:
\[\begin{bmatrix} a & b \ c & d \ e & f \end{bmatrix}\]Here, each letter represents a number or an element of the matrix. It's important to remember the dimensions because they determine what operations you can perform. For instance:

• Addition/Subtraction: A 3x2 matrix can only be added or subtracted from another 3x2 matrix.
• Multiplication: For matrix multiplication, a 3x2 matrix can only multiply a matrix with 2 rows, and the result will be a 3xN matrix (where \( N \) is the number of columns the other matrix has).

Understanding dimensions helps you know which operations are possible and leads to successful calculations in matrix algebra.

Scalar Multiplication

Scalar multiplication is when you multiply every element in a matrix by a single number, known as the scalar. This operation changes each element in the matrix but keeps the matrix's original shape and size.

• Step 1: Identify the scalar. In our example, the scalar is \( 3 \).
• Step 2: Multiply each element individually. For example, if you have a matrix \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \), then multiplying by 3 results in \( \begin{bmatrix} 3 \times 1 & 3 \times 2 \ 3 \times 3 & 3 \times 4 \end{bmatrix} = \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix} \).

Scalar multiplication is straightforward since it only requires basic multiplication. It has no special conditions, making it applicable to any matrix regardless of its size or shape. It's a form of scaling that adjusts the matrix's "size" in terms of value, but not its dimensions.