Math Problem Statement

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Compute the following matrix operation involving matrix multiplication and addition.

Solution

Let's solve the matrix operation step by step.

The given problem involves:

  1. Matrix multiplication
  2. Matrix addition.

We need to compute:

[ \left[ \begin{matrix} -2 & 4 \ 2 & 1 \ 3 & -3 \end{matrix} \right] \cdot \left[ \begin{matrix} -3 & -2 \ 2 & 1 \end{matrix} \right] + \left[ \begin{matrix} 1 & 2 & 3 \ 3 & 4 & 3 \ 5 & 6 & 6 \end{matrix} \right] ]

Step 1: Perform Matrix Multiplication

The first part involves multiplying a 3×23 \times 2 matrix by a 2×22 \times 2 matrix, resulting in a 3×23 \times 2 matrix.

\begin{bmatrix} -2 & 4 \\ 2 & 1 \\ 3 & -3 \end{bmatrix}, \quad \text{Second matrix: } \begin{bmatrix} -3 & -2 \\ 2 & 1 \end{bmatrix}.$$ #### Element-wise calculation of the product: $$\text{Resultant matrix: } \begin{bmatrix} r_{11} & r_{12} \\ r_{21} & r_{22} \\ r_{31} & r_{32} \end{bmatrix}$$ Where: - $$r_{11} = (-2)(-3) + (4)(2) = 6 + 8 = 14$$, - $$r_{12} = (-2)(-2) + (4)(1) = 4 + 4 = 8$$, - $$r_{21} = (2)(-3) + (1)(2) = -6 + 2 = -4$$, - $$r_{22} = (2)(-2) + (1)(1) = -4 + 1 = -3$$, - $$r_{31} = (3)(-3) + (-3)(2) = -9 - 6 = -15$$, - $$r_{32} = (3)(-2) + (-3)(1) = -6 - 3 = -9$$. Thus, the resultant matrix is: $$\begin{bmatrix} 14 & 8 \\ -4 & -3 \\ -15 & -9 \end{bmatrix}.$$ ### Step 2: Add the resultant matrix to the given matrix The next part involves adding the above matrix to: $$\begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 3 \\ 5 & 6 & 6 \end{bmatrix}.$$ Since the dimensions do not match for addition ($$3 \times 2$$ vs. $$3 \times 3$$), there is an error in the formulation of the problem. Matrix addition is only defined for matrices of the same dimensions. Would you like help clarifying this discrepancy or double-checking the problem statement?

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Matrix Addition

Formulas

Matrix multiplication: C[i][j] = sum(A[i][k] * B[k][j]) for all k
Matrix addition: C[i][j] = A[i][j] + B[i][j]

Theorems

Matrix multiplication is defined only if the number of columns in the first matrix equals the number of rows in the second matrix.
Matrix addition is defined only if the matrices have the same dimensions.

Suitable Grade Level

Undergraduate (Year 1-2)