Use the distributive property of matrix multiplication.

Here are the solutions to the problems. 6. Prove that (A+B)(C+D) = AC + AD + BC + BD. Step 1: Use the distributive property of matrix multiplication. Let X = (A+B). Then the expression becomes X(C+D). Applying the left distributive property, X(C+D) = XC + XD. Step 2: Substitute X = (A+B) back into the expression. XC + XD = (A+B)C + (A+B)D Step 3: Apply the right distributive property to each term. (A+B)C = AC + BC (A+B)D = AD + BD Step 4: Combine the results. (A+B)(C+D) = AC + BC + AD + BD Since matrix addition is commutative, the order of terms in a sum does not matter. Therefore, AC + BC + AD + BD = AC + AD + BC + BD. Proven 7. If the matrix A in Example 5 had all its four elements nonzero, would x'Ax still give a weighted sum of squares? Would the associative law still apply? To answer this, we assume x is a column vector, and A is a 2 × 2 matrix. Let x = x_1 \\ x_2 and A = a_11 & a_12 \\ a_21 & a_22 . Then x'Ax = a_11x_1^2 + (a_12+a_21)x_1 x_2 + a_22x_2^2. Would x'Ax still give a weighted sum of squares? If all four elements of A are nonzero, then a_12 and a_21 are nonzero. This means the term (a_12+a_21)x_1 x_2 will generally be present (unless a_12 = -a_21). A "weighted sum of squares" typically refers to a sum of terms of the form w_i x_i^2, without cross-product terms like x_1 x_2. Therefore, if A is not a diagonal matrix (i.e., a_12 or a_21 are nonzero), x'Ax would be a quadratic form but not strictly a "weighted sum of squares" in the sense of only squared terms. ❌ No, it would generally be a quadratic form with cross-product terms, not solely a weighted sum of squares. Would the associative law still apply? The associative law of matrix multiplication, (X'A)X = X'(AX), applies whenever the matrix dimensions are compatible for multiplication. The values of the elements (zero or nonzero) do not affect the validity of the associative law itself. ✅ Yes, the associative law of matrix multiplication always applies when the dimensions are compatible. 8. Name some situations or contexts where the notion of a weighted or unweighted sum of squares may be relevant. Least Squares Regression: In statistics, this method minimizes the unweighted sum of squared residuals* to find the best-fitting line or curve for a set of data points. Weighted Least Squares (WLS): An extension of least squares where different data points are given different importance (weights) in the minimization process, leading to a weighted sum of squared residuals*. This is used when observations have varying precision. Variance and Standard Deviation: These fundamental statistical measures are based on the sum of squared differences* from the mean, representing the spread of data. Error Analysis and Measurement: When combining multiple measurements with different levels of uncertainty, a weighted sum of squares* can be used to determine the most probable value, giving more weight to more precise measurements.