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, $AC + BC + AD + BD = AC + AD + BC + BD$. $\boxed{\text{Proven}}$ 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 $2 \times 2$ matrix. Let $x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ and $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$. Then $x'Ax = a_{11}x_1^2 + (a_{12}+a_{21})x_1 x_2 + a_{22}x_2^2$. 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.