Problem 10 - Algebra and Trigonometry

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

Matrix addition is impossible due to dimension mismatch.

Step 1: Understand Matrix Dimensions

The first matrix has dimensions 3x3, while the second matrix has dimensions 3x2. Matrix addition requires the matrices to be of the same dimensions, which they are not in this case. Thus, addition is not possible.

Step 2: Scalar Multiplication

Given that matrix addition cannot be performed, let us proceed with a possible scalar multiplication as indicated by the multiplication of the first matrix by 2. Calculate this scalar multiplication: \[ 2 \times \begin{bmatrix} 1 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 2 & 0 \ 2 & 0 & 2 \ 0 & 2 & 2 \end{bmatrix} \]

Step 3: Reattempt Addition Reevaluation

Reevaluate matrix addition feasibility. The result from scalar multiplication is a 3x3 matrix, and the other matrix is 3x2; they still do not match in dimensions, confirming matrix addition between them is not possible.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation that requires two matrices to have identical dimensions. This means each matrix must have the same number of rows and columns. To understand why this is important, envision how addition works in simpler arithmetic. We add numbers by pairing each position with its counterpart. Similarly, in matrices, each element in a position of the first matrix must have a matching element in the same position of the second matrix.

• If a matrix is 2x2, it can only be added to another 2x2 matrix. If it's a 3x3 matrix, it needs another 3x3 to add.
• Think of matrix addition as adding two grids of numbers to form one. Each cell of one grid pairs with a cell directly aligned in the other.

If matrices do not share dimensions, addition is impossible. You'll only face errors when mismatched dimensions attempt addition, like trying to fit a square into a rectangle slot.

Scalar Multiplication

Scalar multiplication is another basic matrix operation where each element of a matrix is multiplied by a single number, known as a scalar. This is different from matrix addition because it does not depend on the dimensions of another matrix. Instead, every entry within the matrix changes based on the multiplication with the scalar.

For example, multiplying the matrix \[\begin{bmatrix} 1 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 1\end{bmatrix}\] by a scalar 2 gives: \[2 \times \begin{bmatrix} 1 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 1\end{bmatrix} = \begin{bmatrix} 2 & 2 & 0 \ 2 & 0 & 2 \ 0 & 2 & 2\end{bmatrix}\]

• Every element in the original matrix is simply doubled.
• This operation maintains the original dimensions of the matrix, i.e., a 3x3 remains a 3x3 post-scalar multiplication.
• Scalar multiplication is very handy in scaling matrix data uniformly across its elements.

Matrix Dimensions

Matrix dimensions are critical in determining how matrix operations can be performed. Each matrix is defined by its number of rows and columns, usually denoted as "m x n" where "m" is the number of rows and "n" of columns. Understanding these dimensions will clarify what operations are feasible.

• For matrix addition, both matrices must have identical dimensions. Otherwise, the operation cannot proceed.
• In terms of matrix multiplication, an entirely different operation, it requires that the number of columns in the first matrix equals the number of rows in the second.
• Dimensions give a quick visual guideline: a 3x3 matrix (a square matrix) is very different from a 3x2 matrix (a rectangular matrix). They visually and mathematically belong to separate categories.

In essence, quickly checking dimensions can save potential errors before attempting complex operations. Always inspect and understand matrix sizes before plunging into operations.