Problem 6 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The operation is impossible due to mismatched matrix dimensions.

Step 1: Determine Matrix Dimensions

First, we identify the dimensions of each matrix. The first matrix \( A \) is \( 3 \times 3 \) (3 rows and 3 columns), and the second matrix \( B \) is \( 3 \times 2 \) (3 rows and 2 columns).

Step 2: Verify Matrix Operation Compatibility

We need to verify if these matrices can be added. The addition of matrices requires that they have the same dimensions. Since \( A \) is \( 3 \times 3 \) and \( B \) is \( 3 \times 2 \), they cannot be directly added because their dimensions are not the same.

Step 3: Check for Scalar Multiplication

While adding the matrices isn't possible due to dimension mismatch, we consider other operations stated in the expression. First, we multiply the matrix \( A \) by 2 (scalar multiplication), which changes its elements but not its dimensions.

Step 4: Conclude on the Operation

Since matrix addition requires identical dimensions and our matrices differ in dimensions, combining them via addition is impossible. There's no operation on different-sized matrices given that can result in a valid outcome according to matrix operation rules.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation where two matrices are added together element-by-element. However, this operation is only possible when both matrices have the same dimensions. Each matrix must have the same number of rows and the same number of columns. This ensures that each element in one matrix lines up perfectly with a corresponding element in the other matrix.

For instance, if you have Matrix A of dimensions \(m \times n\) and Matrix B of the same dimensions \(m \times n\), you can add these matrices by simply adding the corresponding elements.

• If Matrix A is \(\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \)
• And Matrix B is \(\begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \)

Then the resulting matrix A + B will be: \[\begin{bmatrix}a_{11} + b_{11} & a_{12} + b_{12}\a_{21} + b_{21} & a_{22} + b_{22}\end{bmatrix}\]If the dimensions do not match, as in the case of exercise, matrix addition cannot proceed.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a single number, known as the scalar. Unlike matrix addition, scalar multiplication is always possible, regardless of the matrix’s size. The operation affects only the values within the matrix, not its overall size or shape.

For example, if you have a matrix \( C \) given as:\[\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\] and you want to multiply it by the scalar \( k = 2 \), the resulting matrix will be:

• Multiply each element of Matrix C by 2:
• \( 2 \times 1 = 2 \)
• \( 2 \times 2 = 4 \)
• \( 2 \times 3 = 6 \)
• \( 2 \times 4 = 8 \)

This gives you:\[\begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}\]In the exercise's case, multiplying the first matrix by 2 simply doubles each of its elements, maintaining its original dimensions of \(3 \times 3\).

Matrix Dimensions

Understanding matrix dimensions is crucial for performing any matrix operations. The dimensions of a matrix are specified by the number of rows followed by the number of columns it contains. For example, a matrix with 3 rows and 2 columns is denoted as a \(3 \times 2\) matrix. These dimensions are critical when determining the feasibility of operations like addition, multiplication, and others.

Matrix operations such as addition and multiplication have strict rules regarding dimensions.

• For addition, matrices must have identical dimensions.
• For multiplication, the number of columns in the first matrix must equal the number of rows in the second.

In the given exercise, the first matrix is \(3 \times 3\) and the second is \(3 \times 2\), making direct addition impossible. Keeping an eye on these dimensions will guide you in deciding what matrix operations you can perform. Understanding this allows you to efficiently tackle matrix-based problems with ease.