📘 Linear Algebra

1. **Problem (a):** Prove that if $A$ and $B$ are symmetric $n \times n$ matrices, then $A + cB$ is symmetric for all scalars $c$. 2. **Step 1:** Recall the definition of a symmetr

1. The problem states that vectors $\mathbf{a}$ and $\mathbf{b}$ are orthogonal, meaning their dot product is zero: $$\mathbf{a} \cdot \mathbf{b} = 0.$$\n\n2. We are given the magn

1. **State the problem:** We have a system of linear equations: $$\begin{cases} 5x_1 + 2x_2 + 5x_3 + 2x_4 = 2 \\ -20x_1 - 3x_2 - 13x_3 - 8x_4 = -8 \\ -15x_1 - 6x_2 - 15x_3 - 6x_4 =

1. **State the problem:** We have a system of linear equations and need to analyze it using Kronecker's Rank Theorem. 2. **Given system:**

1. The problem is to find matrices $M$ and $N$ that are maximal square submatrices of a given matrix $C$ with nonzero determinants. 2. A maximal square submatrix means the largest

1. **State the problem:** We have a system of linear equations: $$\begin{cases} 5x_1 + 2x_2 + 5x_3 + 2x_4 = 2 \\ -20x_1 - 3x_2 - 13x_3 - 8x_4 = -8 \\ -15x_1 - 6x_2 - 15x_3 - 6x_4 =

1. **State the problem:** We have a system of three linear equations with four variables $x_1, x_2, x_3, x_4$: $$\begin{cases} 3x_1 + 2x_2 + 2x_3 + 4x_4 = 0 \\ -12x_1 - 4x_2 - 10x_

1. The problem states: Make matrix $M$ be a square matrix of $C$ and matrix $N$ a square submatrix of $A$. 2. To clarify, a square matrix is a matrix with the same number of rows a

1. **State the problem:** We are given a system of linear equations and asked to analyze it using Kronecker's Rank Theorem. 2. **Part a:** Find the rank of the coefficient matrix $

1. **State the problem:** We have a system of linear equations: $$\begin{cases} 3x_1 + 2x_2 + 2x_3 + 4x_4 = 0 \\ -12x_7 - 4x_7 - 10x_3 - 14x_4 = 5 \\ -9x_1 - 6x_7 - 9x_3 - 12x_4 =

1. **Solve the system** given by the augmented matrix \(\begin{bmatrix}1 & 3 & 1 & 0 \\ 2 & 6 & 4 & 8 \\ 0 & 0 & 2 & 4\end{bmatrix} \begin{bmatrix}x \\ y \\ z \\ t\end{bmatrix} = \

1. The problem asks us to prove that for a set $C$, the double orthogonal complement $C^{\perp\perp}$ equals $C$. 2. Recall the definition: For a set $C$ in an inner product space,

1. **State the problem:** We are given vectors \(u = \begin{bmatrix}1 \\ -1 \\ 0\end{bmatrix}\), \(v = \begin{bmatrix}-2 \\ 1 \\ 1\end{bmatrix}\), \(w = \begin{bmatrix}1 \\ 2 \\ 3\

1. **State the problem:** Find the determinant of the 3x3 matrix $$\begin{bmatrix} 3 & -1 & 1 \\ -1 & 7 & -2 \\ 2 & 6 & 1 \end{bmatrix}$$

1. **State the problem:** Find the eigenvectors of the matrix $$A = \begin{pmatrix}1 & 1 \\ 3 & -1\end{pmatrix}$$ using the transformation method. 2. **Find the eigenvalues:** Solv

1. **Problem 13 (Leslie Model Population Growth):** Given Leslie matrix:

1. **Find the eigenvalues and eigenvectors of matrix** $$A=\begin{bmatrix}1 & 1 & 3 \\ 3 & 5 & 1 \\ 3 & 1 & 1\end{bmatrix}$$

1. The problem is to verify the vector addition: \((3, 2) = (1, 0) + (2, 2)\). 2. Vector addition means adding corresponding components:

1. **State the problem:** Find the eigenvalues of the matrix $$A = \begin{bmatrix} 6 & -2 & -2 \\ -2 & 3 & -1 \\ 2 & 1 & 3 \end{bmatrix}$$. 2. **Recall the characteristic equation:

1. **State the problem:** We want to transform the quadratic form $$3x_1^2 + 3x_2^2 - 5x_3^2 - 2x_1x_2 - 6x_3x_2 - 6x_1x_3$$ into its canonical form using an orthogonal transformat

1. **State the problem:** Solve for matrix $X$ in the equation $$\begin{bmatrix}3 & -4 \\ 2 & -5\end{bmatrix} \left(X - \begin{bmatrix}2 & 0 & 3\\0 & 1 & -1\end{bmatrix} \cdot \beg