Problem 47 - Algebra 2

Question

Find each product, if possible. \(\left[\begin{array}{rr}{5} & {-4} \\ {8} & {3}\end{array}\right] \cdot\left[\begin{array}{l}{5} \\ {1}\end{array}\right]\)

Step-by-Step Solution

Answer

The product is \( \left[ \begin{array}{r} 21 \\ 43 \end{array} \right] \).

Step 1: Check Matrix Multiplication Validity

To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The first matrix is a 2x2 matrix and the second matrix is a 2x1 matrix, so multiplication is possible.

Step 2: Multiply Row 1 by Column 1

For the first element of the product matrix, we multiply the first row of the first matrix by the column of the second matrix: \( (5 \times 5) + (-4 \times 1) = 25 - 4 = 21 \).

Step 3: Multiply Row 2 by Column 1

For the second element of the product matrix, multiply the second row of the first matrix by the column of the second matrix: \( (8 \times 5) + (3 \times 1) = 40 + 3 = 43 \).

Step 4: Write the Resultant Matrix

Combine the results to write the product matrix: \( \left[ \begin{array}{r} 21 \ 43 \end{array} \right] \).

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple yet powerful tool in mathematics. It consists of two rows and two columns, where each element of the matrix is carefully positioned to hold a specific numerical value.
The structure of a 2x2 matrix looks like this:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]where \(a, b, c,\) and \(d\) represent numerical entries.
Each element in the matrix will play a role in mathematical operations, particularly in multiplication with other matrices or vectors.
2x2 matrices are an essential part of linear algebra, often used to simplify complex problems into manageable calculations and to represent linear transformations.

2x1 Matrix

The 2x1 matrix, also known as a column vector, is a smaller matrix that contains two rows and one column. Its structure is straightforward:\[\begin{bmatrix} x \ y \end{bmatrix}\]here, \(x\) and \(y\) are the elements of the vector.
A 2x1 matrix can represent points in space or simple vectors in physics.
When multiplied by a 2x2 matrix, the result will be a new 2x1 matrix, which can show a transformation of the original vector.

Matrix Product

The matrix product is the outcome of multiplying two matrices together following specific rules.
To determine if two matrices can be multiplied, check that the number of columns in the first matrix matches the number of rows in the second matrix.
This condition ensures that every element in the first matrix finds a corresponding element in the second matrix.
When multiplying matrices, each element of the resulting matrix is a sum of products. In our example:

• Multiply the first row of the 2x2 matrix with the column of the 2x1 matrix.
• Repeat the process with the second row.

The results form the new matrix, encapsulating the transformed information.

Scalar Multiplication

Scalar multiplication involves multiplying each element of the matrix by a constant value, called the scalar. Although not directly applied in our example, understanding this concept helps in various calculations and transformations in linear algebra.
In scalar multiplication:

• Multiply every element in a matrix by the scalar value.
• The final result is a matrix of the same size but with scaled values.

For example, if a scalar of 3 multiplies a matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the resultant matrix would be \(\begin{bmatrix} 3a & 3b \ 3c & 3d \end{bmatrix}\).
This operation is crucial in fields like physics and engineering, where scaling of values is required for simulations or real-world models.