Mathematics II · Unit 3 · 5 hrs
Matrix Algebra
Exam-focused notes for Matrix Algebra (Mathematics II, MTH168): what the TU syllabus asks and how it has actually been tested, with 10 solved past questions from this unit.
What this unit covers
- Matrix operations
- The inverse of a matrix
- Characterizations of invertible matrices
- Partitioned matrices
- Matrix factorization
- The Leontief input output model
- Subspace of Rn
- Dimension and rank
Partitioned matrices
Find AB by block multiplication of the matrices.
$$A = \begin{bmatrix} 2 & -3 & -1 & 0 & -4 \ 1 & -5 & -2 & 3 & -1\ 0 & -4 & -2 & 7 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} 6 & 4 \ 2 & -1 \ -3 & 7 \ 1 & 3 \ 5 & -3 \end{bmatrix}$$
[10]
$$A = \begin{bmatrix} 2 & -3 & -1 & 0 & -4 \\ 1 & -5 & -2 & 3 & -1\\ 0 & -4 & -2 & 7 & -1 \end{bmatrix}{3\times5}, \quad B = \begin{bmatrix} 6 & 4 \\ 2 & -1 \\ -3 & 7 \\ 1 & 3 \\ 5 & -3 \end{bmatrix}{5\times2}$$ Result $AB$ will be $3\times 2$. Split A colu...
Full solved answer →Matrix operations
Let $A = \begin{bmatrix} 1 & 5 \ -3 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 4 & -5 \ 3 & k \end{bmatrix}$
What value(s) of k, if any, will make AB = BA? [5]
$$A = \begin{bmatrix} 1 & 5 \\ -3 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix}$$ We must find $k$ (if any) so that $AB = BA$. --- $$AB = \begin{bmatrix} 1 & 5 \\ -3 & 1 \end{bmatrix} \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix...
Full solved answer →Let A and B be matrices. Determine the value(s) of k if any will make AB = BA.
$$A = \begin{bmatrix} 2 & 5 \ -3 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & -5 \ 3 & k \end{bmatrix}$$
[5]
$$A = \begin{bmatrix} 2 & 5 \\ -3 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix}$$ Requirement: find $k$ (if any) such that $AB = BA$. $$AB = \begin{bmatrix} 2 & 5 \\ -3 & 1 \end{bmatrix}\begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatr...
Full solved answer →Let A and B be matrices. What value(s) of k if any will make AB = BA?
$$A = \begin{bmatrix} -1 & -2 \ 5 & 9 \end{bmatrix}, \quad B = \begin{bmatrix} 9 & 2 \ k & -1 \end{bmatrix}$$
[5]
$$A = \begin{bmatrix} -1 & -2 \\ 5 & 9 \end{bmatrix}, \quad B = \begin{bmatrix} 9 & 2 \\ k & -1 \end{bmatrix}$$ Requirement: $AB = BA$. $$AB = \begin{bmatrix} (-1)(9)+(-2)(k) & (-1)(2)+(-2)(-1) \\ (5)(9)+(9)(k) & (5)(2)+(9)(-1) \end{bmatrix} = \begin{bmatri...
Full solved answer →The Leontief input output model
The economy whose consumption matrix C is and the final demand is 50 units for manufacturing, 30 units for agriculture and 20 units for service. Find the production level x that will satisfy this demand.
$$C = \begin{bmatrix} 0.5 & 0.4 & 0.2 \ 0.2 & 0.3 & 0.1 \ 0.1 & 0.1 & 0.3 \end{bmatrix}$$
[10]
- Consumption matrix: $$C = \begin{bmatrix} 0.5 & 0.4 & 0.2 \\ 0.2 & 0.3 & 0.1 \\ 0.1 & 0.1 & 0.3 \end{bmatrix}$$ - Final demand: $d = \begin{bmatrix} 50 \\ 30 \\ 20 \end{bmatrix}$ We solve $(I - C)x = d$. $$I - C = \begin{bmatrix} 0.5 & -0.4 & -0.2 \\ -0.2...
Full solved answer →Matrix factorization
Find the LU factorization of $$\begin{bmatrix} 2 & 4 & -1 & 5 & -2 \ -4 & -5 & 3 & -8 & 1 \ 2 & -5 & -4 & 1 & 8 \ -6 & 0 & 7 & -3 & 1 \end{bmatrix}$$ [10]
$$A = \begin{bmatrix} 2 & 4 & -1 & 5 & -2 \\ -4 & -5 & 3 & -8 & 1 \\ 2 & -5 & -4 & 1 & 8 \\ -6 & 0 & 7 & -3 & 1 \end{bmatrix}$$ We seek $A = LU$ with $L$ unit lower triangular ($4\times4$) and $U$ upper (echelon) form ($4\times5$). The multipliers used in G...
Full solved answer →Find LU factorization of the matrix $\begin{bmatrix} 2 & 5 \ 6 & -7 \end{bmatrix}$ [5]
$$A = \begin{bmatrix} 2 & 5 \\ 6 & -7 \end{bmatrix}$$ Goal: find $L$ (lower triangular, unit diagonal) and $U$ (upper triangular) such that $A = LU$. --- Pivot (1,1) = 2. Eliminate entry $a{21} = 6$. Multiplier: $$\ell{21} = \frac{6}{2} = 3$$ Row operation:...
Full solved answer →Find LU Factorization. Given the matrix: $$\begin{bmatrix} 2 & 3 & 4 \ 4 & 5 & 10 \ 4 & 8 & 2 \end{bmatrix}$$ [5]
$$A = \begin{bmatrix} 2 & 3 & 4 \\ 4 & 5 & 10 \\ 4 & 8 & 2 \end{bmatrix}$$ Goal: find $L$ (unit lower triangular) and $U$ (upper triangular) with $A = LU$ using Doolittle's method. --- Pivot $= 2$. Multiplier $m{21} = \frac{4}{2} = 2$: $$R2 \to R2 - 2R1 = [...
Full solved answer →The inverse of a matrix
What is the condition of a matrix to have an inverse? Find the inverse of the matrix if it exists.
$$A = \begin{bmatrix} 5 & 1 & 2 \ 1 & 0 & 3 \ 4 & -3 & 8 \end{bmatrix}$$
[10]
$$A = \begin{bmatrix} 5 & 1 & 2 \\ 1 & 0 & 3 \\ 4 & -3 & 8 \end{bmatrix}$$ A square matrix $A$ has an inverse if and only if it is non-singular, i.e. $\det(A) \neq 0$. Then: $$A^{-1} = \frac{1}{\det(A)}\,\text{adj}(A)$$ where $\text{adj}(A)$ is the transpos...
Full solved answer →What is the condition of a matrix to have an inverse? Find the inverse of the matrix $A = \begin{bmatrix} 5 & 1 & 2 \ 1 & 0 & 3 \ 4 & -3 & 8 \end{bmatrix}$ [10]
$$A = \begin{bmatrix} 5 & 1 & 2 \\ 1 & 0 & 3 \\ 4 & -3 & 8 \end{bmatrix}$$ Required: (a) condition for existence of inverse, (b) find $A^{-1}$. A square matrix $A$ of order $n \times n$ has an inverse if and only if it is non-singular, that is: $$\det(A) \n...
Full solved answer →Make Unit 3 stick
Practice MTH168 with flashcards & quizzes