Problem 48 - Algebra 2

Question

Perform the indicated matrix operations. If the matrix does not exist, write impossible. \(\left[\begin{array}{lll}{3} & {5} & {9}\end{array}\right]+\left[\begin{array}{l}{5} \\ {2} \\\ {6}\end{array}\right]\)

Step-by-Step Solution

Answer

Impossible, matrices have different dimensions.

Step 1: Identify Matrix Compatibility

First, examine the dimensions of the matrices to determine if they can be added. The first matrix is a 1x3 row vector, while the second matrix is a 3x1 column vector.

Step 2: Determine Feasibility of Addition

In matrix addition, both matrices must have the same dimensions. Since one matrix is 1x3 and the other is 3x1, they do not have the same dimensions and cannot be added.

Step 3: Conclusion: Matrix Addition

Since the matrices have incompatible dimensions for addition, the operation is impossible.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. It involves adding two matrices element-wise, which means corresponding elements of each matrix are added together to generate a new matrix. For this addition to be possible, it's crucial that both matrices have the same dimensions. This is because each element in one matrix needs a partner element in the other matrix to result in corresponding sums.
For example, if both matrices are 2x2 like: \(\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}\)and \(\begin{bmatrix} 5 & 7 \ 6 & 8 \end{bmatrix}\)you would add them to get \(\begin{bmatrix} 1+5 & 3+7 \ 2+6 & 4+8 \end{bmatrix}\)which simplifies to \(\begin{bmatrix} 6 & 10 \ 8 & 12 \end{bmatrix}\). Matrix addition is not feasible when matrices don't match in size.

Matrix Dimensions

Matrix dimensions are a way to describe the size of a matrix. A matrix is usually defined by the number of its rows and columns, often given in the form of 'row x column'. This is expressed as \(m \times n\) where \(m\) represents the number of rows and \(n\) the number of columns.
Understanding matrix dimensions is crucial when performing operations such as addition, subtraction, or multiplication in matrices.

• A 3x2 matrix has 3 rows and 2 columns.
• A 1x3 matrix, also known as a row vector, has 1 row and 3 columns.
• A 4x1 matrix, or a column vector, contains 4 rows and 1 column.

Mismatched dimensions often indicate that certain matrix operations cannot proceed, as seen in the original exercise where different dimensions rendered addition impossible.

Incompatible Matrices

Incompatible matrices are matrices that cannot undergo certain operations together due to mismatched dimensions or other restrictions. In the context of matrix addition, the primary requirement is that matrices must be of the same size. If they aren't, they're called incompatible matrices for addition.
Here’s a straightforward example of such incompatibility. Consider a 2x3 matrix and a 3x2 matrix. Their dimensions do not match, thus preventing them from being added.

• For matrix addition: both matrices must have the same dimensional size—let this be a firm rule.
• If a row vector (like 1x3) tries to add with a column vector (like 3x1), this leads to an impossible situation.

Identifying such incompatibilities is a critical step in matrix operations, similar to checking dimensions before addition. By recognizing incompatible matrices early, one can save time and effort in mathematical problem-solving.