# Recent questions in Matrix transformations

Matrix transformations

### For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A. $$A=\begin{bmatrix}2&3&5&-9\\-8&-9&-11&21\\4&-3&-17&27\end{bmatrix}$$ Find a nonzero vector in Nul A. $$A=\begin{bmatrix}-3\\2\\0\\1\end{bmatrix}$$

Matrix transformations

### Find k such that the following matrix M is singular. $$M=\begin{bmatrix}-1 & -1 & -2\\ 0 & -1 & -4 \\ -12+k & -2 & -2 \end{bmatrix}$$ $$k=?$$

Matrix transformations

### Use the given inverse of the coefficient matrix to solve the following system $$5x_1+3x_2=6$$ $$-6x_1-3x_2=-2$$ $$A^{-1}=\begin{bmatrix}-1 & -1 \\2 & \frac{5}{3} \end{bmatrix}$$

Matrix transformations

### Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. $$\begin{bmatrix}2 & 5&5 \\5 & 2&5\\5&5&2 \end{bmatrix}\lambda=-3.12$$ Find P and D

Matrix transformations

### Give the elementary matrix that converts $$\begin{bmatrix} -2 & -2 & -1 \\ -3 & -1 & -3 \\ 1 & -4 & -3 \end{bmatrix}$$ to $$\begin{bmatrix} -6 & -2 & -1 \\ -5 & -1 & -3 \\ -7 & -4 & -3 \end{bmatrix}$$

Matrix transformations

### Let T be the linear transformation from R2 to R2 consisting of reflection in the y-axis. Let S be the linear transformation from R2 to R2 consisting of clockwise rotation of $$30^{\circ}$$. (b) Find the standard matrix of $$T , [T ]$$. If you are not sure what this is, see p. 216 and more generally section 3.6 of your text. Do that before you go looking for help!

Matrix transformations

### Proove that the set of oil $$2 \times 2$$ matrices with entries from R and determinant +1 is a group under multiplication

Matrix transformations

### (7) If A and B are a square matrix of the same order. Prove that $$(ABA^{−1})^3=AB^3A^{−1}$$

Matrix transformations

### If $$A =[1,2,4,3]$$, find B such that $$A+B=0$$

Matrix transformations

### Write the given matrix equation as a system of linear equations without matrices. $$\displaystyle{\left[\begin{matrix}{2}&{0}&-{1}\\{0}&{3}&{0}\\{1}&{1}&{0}\end{matrix}\right]}{\left[\begin{matrix}{x}\\{y}\\{z}\end{matrix}\right]}={\left[\begin{matrix}{6}\\{9}\\{5}\end{matrix}\right]}$$

Matrix transformations

### Let A be a $$6 \times 9$$ matrix. If Nullity $$\displaystyle{\left({A}{T}{A}^{{T}}{A}{T}\right)}$$ $$= 2$$ then Nullity $$(A) = 2$$

Matrix transformations

### Write an augumented matrix for the system of linear equations $$\displaystyle{\left[\begin{matrix}{x}-{y}+{z}={8}\\{y}-{12}{z}=-{15}\\{z}={1}\end{matrix}\right]}$$

Matrix transformations

### 1:Find the determinant of the following mattrix $$\displaystyle{\left[\begin{matrix}\begin{matrix}{2}&-{1}&-{6}\end{matrix}\\\begin{matrix}-{3}&{0}&{5}\end{matrix}\\\begin{matrix}{4}&{3}&{0}\end{matrix}\end{matrix}\right]}$$ 2: If told that matrix A is singular Matrix find the possible value(s) for x $$\displaystyle{A}={\left\lbrace\begin{matrix}{16}{x}&{4}{x}\\{x}&{9} \end{matrix}\right.}$$

Matrix transformations

### Let D be the diagonal subset $$\displaystyle{D}={\left\lbrace{\left({x},{x}\right)}{\mid}{x}∈{S}_{{3}}\right\rbrace}$$ of the direct product $$S_3 \times S_3$$. Prove that D is a subgroup of $$S_3 \times S_3$$ but not a normal subgroup.

Matrix transformations

### Show that W, the set of all $$3 \times 3$$ upper triangular matrices, forms a subspace of all $$3 \times 3$$ matrices. What is the dimension of W? Find a basis for W.

Matrix transformations

### It can be shown that the algebraic multiplicity of an eigenvalue lambda is always greater than or equal to the dimension of the eigenspace corresponding to lambda. Find h in the matrix A below such that the eigenspace for lambda = 5 is two-dimensional: $$\displaystyle{A}={\left[\begin{array}{cccc} {5}&-{2}&{6}&-{1}\\{0}&{3}&{h}&{0}\\{0}&{0}&{5}&{4}\\{0}&{0}&{0}&{1}\end{array}\right]}$$

Matrix transformations

### Find basis and dimension $$\displaystyle{\left\lbrace{x}{e}{R}^{{4}}{\mid}{x}{A}={0}\right\rbrace}$$ where $$\displaystyle{A}={\left[-{1},{1},{2},{1},{1},{0},{2},{3}\right]}^{{T}}$$

Matrix transformations

### Let $$\displaystyle{u}={i}+{2}{j}-{3}{k}$$ and $$\displaystyle{v}={2}{i}+{3}{j}+{k}\in{R}^{{3}}$$ (a) What is $$u \cdot v$$? (b) What is $$u \cdot v$$?

Matrix transformations