Problem 25 - Algebra 2

Question

Perform the indicated matrix operation. If the matrix does not exist, write impossible. $$ -2\left[\begin{array}{rrr}{2} & {-4} & {1} \\ {-3} & {5} & {8} \\ {7} & {6} & {-2}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} -4 & 8 & -2 \\ 6 & -10 & -16 \\ -14 & -12 & 4 \end{bmatrix} \).

Step 1: Understand the Matrix

The matrix operation requires us to multiply the scalar (-2) with each element of the given matrix. The matrix provided is a 3x3 matrix: $$ \begin{bmatrix} 2 & -4 & 1 \ -3 & 5 & 8 \ 7 & 6 & -2 \end{bmatrix} $$

Step 2: Multiply by the Scalar

Multiply every element of the matrix by (-2). This involves simple multiplication for each entry of the matrix.

Step 3: Calculate Each Entry

Carry out the multiplication for each element: - (-2) * 2 = -4 - (-2) * (-4) = 8 - (-2) * 1 = -2 - (-2) * (-3) = 6 - (-2) * 5 = -10 - (-2) * 8 = -16 - (-2) * 7 = -14 - (-2) * 6 = -12 - (-2) * (-2) = 4

Step 4: Assemble the Resulting Matrix

The resulting matrix after performing the scalar multiplication is:$$ \begin{bmatrix} -4 & 8 & -2 \ 6 & -10 & -16 \ -14 & -12 & 4 \end{bmatrix} $$

Key Concepts

Scalar Multiplication

In the world of mathematics, scalar multiplication is a simple yet important operation that involves multiplying every element of a matrix by a constant, known as a scalar. For example, if we have a scalar, say \(-2\), and a matrix, we need to multiply \(-2\) with each number inside the matrix. This action stretches or compresses the matrix values, depending on the scalar.
To visualize, take the matrix:

• 2, -4, 1
• -3, 5, 8
• 7, 6, -2

With \(-2\) as our scalar, multiply each number. The steps are straightforward with the result being a new matrix where each element has been transformed by this multiplication. This process is essential for manipulating matrices in various mathematical applications like physics and engineering.

3x3 Matrix

A 3x3 matrix is a square matrix that has 3 rows and 3 columns. Think of it as a small grid filled with numbers. Each grid position holds a number, and these are systematically arranged in rows (horizontal) and columns (vertical).
A convenient way to represent a 3x3 matrix is:

• Row 1: 2, -4, 1
• Row 2: -3, 5, 8
• Row 3: 7, 6, -2

Having equal numbers of rows and columns, a 3x3 matrix is an important construct in linear algebra. It's commonly used in transformation processes, graphics, and systems of linear equations. You can add, subtract, or even multiply these matrices to achieve different outcomes. Each operation adheres to certain rules specific to matrices.

Matrix Multiplication

Matrix multiplication is more complex than multiplying regular numbers. It involves the rows of the first matrix and the columns of the second. But here's a key rule: the number of columns in the first matrix must equal the number of rows in the second for it to be possible.
When dealing with a 3x3 matrix multiplied by another 3x3 matrix, the process becomes intricate. You take each element of a row from the first matrix, multiply them by corresponding elements in a column of the second matrix, and sum the products. This gives one entry in a new resulting matrix.

• First Matrix Row 1 ⤳ Second Matrix Column 1 = Result Element (Row 1, Column 1)
• First Matrix Row 2 ⤳ Second Matrix Column 2 = Result Element (Row 2, Column 2)
• And so on for the rest...

Through matrix multiplication, you can manipulate data, solve complex equations, and even transform spaces in graphical applications, making it a powerful tool in various scientific fields.