Problem 12 - Precalculus: Mathematics 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 matrix operation is impossible due to mismatched matrix dimensions.

Step 1: Matrix Multiplication by a Scalar

Multiply each element of the first matrix by the scalar 2. The matrix \[\left[\begin{array}{ccc} 1 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 1 \end{array}\right]\] becomes \[\left[\begin{array}{ccc} 2 & 2 & 0 \ 2 & 0 & 2 \ 0 & 2 & 2 \end{array}\right].\]

Step 2: Check Matrix Dimensions for Addition

Compare the dimensions of the two matrices involved in addition. The first matrix, after scalar multiplication, is 3x3 and the second matrix is 3x2. Verify if matrix addition is possible by checking if the dimensions match.

Step 3: Conclusion - Addition Impossible

Matrix addition requires both matrices to have the same dimensions. Since the resulting matrix from the first operation is 3x3 and the other matrix is 3x2, they cannot be added because their dimensions do not align.

Key Concepts

Matrix Addition

Matrix addition is a straightforward operation, but it requires that both matrices have the same dimensions. This means they must have the same number of rows and columns. When you add matrices, you simply add each corresponding element from the two matrices together.
For example, consider two matrices of the same size \(A = \left[\begin{array}{cc} 1 & 2 \ 3 & 4 \end{array}\right],B = \left[\begin{array}{cc} 5 & 6 \ 7 & 8 \end{array}\right]\).The sum \(A + B\)will be calculated as follows: \(\left[\begin{array}{cc} 1+5 & 2+6 \ 3+7 & 4+8 \end{array}\right] = \left[\begin{array}{cc} 6 & 8 \ 10 & 12 \end{array}\right]\).However, if the matrices do not share the same dimensions, as in the given exercise with the after-scalar-multiplication resulting in a \(3\times3\) and attempting to add it to a \(3\times2\) matrix, matrix addition cannot be performed.
It is crucial always to check dimensions before attempting this operation.

Matrix Multiplication

Matrix multiplication is a bit more complex than addition and requires a specific set of conditions for it to be possible. To multiply two matrices \(A \)and \(B \),you must ensure that the number of columns in \(A \)is equal to the number of rows in \(B \).The resulting product will be a new matrix with dimensions equal to the number of rows of \(A \)and the number of columns of \(B \).
Here's how it typically works: each element in the resulting matrix is a dot product of a row from \(A \)and a column from \(B \). For example, if \(A = \left[\begin{array}{cc} 1 & 2 \ 3 & 4 \end{array}\right]\)and \(B = \left[\begin{array}{cc} 5 & 6 \ 7 & 8 \end{array}\right]\),the product \(AB \)will be \(\left[\begin{array}{cc} 1\times5 + 2\times7 & 1\times6 + 2\times8 \ 3\times5 + 4\times7 & 3\times6 + 4\times8 \end{array}\right] = \left[\begin{array}{cc} 19 & 22 \ 43 & 50 \end{array}\right]\).If either condition - the matching columns and rows - is not met, then matrix multiplication cannot proceed.

Scalar Multiplication

Scalar multiplication is the simplest form of matrix operation. It involves multiplying each element of a matrix by a single number, called a scalar. This is similar to distributing a constant across each term in an algebraic expression.
Take a matrix \(A = \left[\begin{array}{cc} 1 & 2 \ 3 & 4 \end{array}\right]\)and a scalar \(3\).The result of the scalar multiplication will be \(3 \cdot A = \left[\begin{array}{cc} 3 \times 1 & 3 \times 2 \ 3 \times 3 & 3 \times 4 \end{array}\right] = \left[\begin{array}{cc} 3 & 6 \ 9 & 12 \end{array}\right]\).This operation is straightforward and does not require any special conditions other than applying the scalar evenly across all elements.
It's important because it can be used to adjust matrices to be compatible with others during more complex operations.