1. a) The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. b) Given matrix $$A=\begin{bmatrix}8 & 1 & 3 & 6 \\ 0 & 3 & 2 & 2 \\ -8 & -1 & -3 & 4\end{bmatrix}$$ Using row operations we check if rows are independent. The matrix has 3 rows and 4 columns. c) Matrix $$B = \begin{bmatrix}1 & 3 & 4 \\ 3 & 2 & 5 \\ 2 & 0 & 3 \end{bmatrix}$$ Step 1: Subtract 3 * Row 1 from Row 2: $$R_2 = R_2 - 3R_1 \Rightarrow \begin{bmatrix}3 & 2 & 5 \end{bmatrix} - 3 \times \begin{bmatrix}1 & 3 & 4\end{bmatrix} = \begin{bmatrix}0 & -7 & -7 \end{bmatrix}$$ Step 2: Subtract 2 * Row 1 from Row 3: $$R_3 = R_3 - 2R_1 = \begin{bmatrix}0 & -6 & -5\end{bmatrix}$$ Step 3: Make leading coefficient of Row 2 as 1: divide by -7 $$R_2 = \begin{bmatrix}0 & 1 & 1\end{bmatrix}$$ Step 4: Eliminate the 2nd element in Row 3: $$R_3 = R_3 - (-6)R_2 = \begin{bmatrix}0 & 0 & 1\end{bmatrix}$$ Resulting Echelon form: $$\begin{bmatrix}1 & 3 & 4 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{bmatrix}$$ Rank is number of non-zero rows: 3. 2. a) System: $$4x - 3y - 9z + 6w = 0 \\ 2x + 3y + 3z + 6w = 6 \\ 4x - 21y - 39z - 6w = -24$$ Form augmented matrix and reduce to check consistency, obtain solutions. b) System: $$x + 2y + 2z = 2; \\ 3x - 2y - z = 5; \\ 2x - 5y + 3z = -4; \\ x + 4y + 6z = 0$$ Check for consistency by comparing rank of coefficient matrix and augmented matrix. c) System: $$x + y + z = 3 \\ x + 2y + 2z = 6 \\ x + \lambda y + 3z = \mu$$ (i) No solution occurs if the augmented matrix rank is greater than coefficient matrix rank. (ii) Unique solution if ranks are equal and equal to number of variables. (iii) Infinite solutions if ranks equal but less than number of variables. 3. a) Eigenvalues are scalars $\lambda$ such that $Ax = \lambda x$ for some non-zero vector $x$ called eigenvector. b) Matrix $$M = \begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 3 & 3 & 1 \end{bmatrix}$$ Sum of eigenvalues = trace = sum of diagonal entries = $1 + 1 + 1 = 3$. Product of eigenvalues = determinant of matrix. c) For matrix $$\begin{bmatrix}5 & 7 & 3 \\ -2 & k & 5 \\ 0 & 3 & 2 \end{bmatrix}$$ Sum of eigenvalues = trace = $5 + k + 2 = -10$ implies $k = -17$. 4. a) Matrix $$\begin{bmatrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & -1 & 1 \end{bmatrix}$$ Find eigenvalues by solving characteristic polynomial and corresponding eigenvectors by solving $(A-\lambda I)x=0$. b) Matrix $$A=\begin{bmatrix}11 & -4 & -7 \\ 7 & -2 & -5 \\ 10 & -4 & -6 \end{bmatrix}$$ Find eigenvalues and eigenvectors similarly. 5. a) Cayley-Hamilton theorem states every square matrix satisfies its own characteristic equation. b) For matrix $$A=\begin{bmatrix}1 & 4 \\ 3 & 2 \end{bmatrix}$$ Find characteristic polynomial, verify $f(A)=0$ and find inverse using theorem. c) For matrix $$A=\begin{bmatrix}3 & 1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 5 \end{bmatrix}$$ Use characteristic polynomial and Cayley-Hamilton theorem to compute $A^{-1}$. d) For matrix $$A=\begin{bmatrix}1 & 2 \\ 2 & -1 \end{bmatrix}$$ Use Cayley-Hamilton theorem to find $A^8$. 6. a) The quadratic form $x^2 - 3z^2 + 2xy - 3yz$ corresponds to symmetric matrix $$Q = \begin{bmatrix}1 & 1 & 0 \\ 1 & 0 & -\frac{3}{2} \\ 0 & -\frac{3}{2} & -3\end{bmatrix}$$ b) Quadratic form $Q=2(x^2 + xy + y^2)$ has matrix $$2\begin{bmatrix}1 & \frac{1}{2} \\ \frac{1}{2} & 1\end{bmatrix} = \begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix}$$ c) Quadratic form related to $$\begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 3 & 3 & 1\end{bmatrix}$$ is $x^2 + 4xy + 6xz + y^2 + 6yz + z^2$. 7. a) Given quadratic form $$6x_1^2 + 3x_2^2 + 3x_3^2 - 4x_1x_2 - 2x_2x_3 + 4x_1x_3$$ Reduce it using orthogonal transformations to canonical form with no cross terms. b) For quadratic form $$3x^2 + 3y^2 + 3z^2 + 2xy - 2yz + 2zx$$ Similarly, diagonalize to get canonical form.