A 2-digit number has two properties: The digits sum to 11, and if the number is written with digits reversed, and subtracted from the original number, the result is 45.

The key to this problem is noticing that our 2-digit number can be written as $10A + B$, where $A, B$ are the 10s and 1s digit respectively.

The digits summing to 11 then yields the equality $A + B = 11$.

The number with digits reversed is $10B + A$, and so the second property yields the equation $10A + B – (10B + A) = 45$. Simplifying, we have the system of equations
\begin{align*}
A+B &= 11\\
9A-9B &= 45
\end{align*}

To solve this system, we create the augmented matrix and use elementary row operations to put it into reduced row-echelon form:
\begin{align*}
\left[\begin{array}{rr|r} 1 & 1 & 11 \\ 9 & -9 & 45 \end{array} \right] \xrightarrow{ R_2 – 9 R_1 } \left[\begin{array}{rr|r} 1 & 1 & 11 \\ 0 & -18 & -54 \end{array} \right]\\[6pt] \xrightarrow{ \frac{-1}{18} R_2 } \left[\begin{array}{rr|r} 1 & 1 & 11 \\ 0 & 1 & 3 \end{array} \right] \xrightarrow{R_1 – R_2} \left[\begin{array}{rr|r} 1 & 0 & 8 \\ 0 & 1 & 3 \end{array} \right].
\end{align*}
With this final augmented matrix, we can read off the solution $A = 8$ and $B = 3$.

A Condition that a Linear System has Nontrivial Solutions
For what value(s) of $a$ does the system have nontrivial solutions?
\begin{align*}
&x_1+2x_2+x_3=0\\
&-x_1-x_2+x_3=0\\
& 3x_1+4x_2+ax_3=0.
\end{align*}
Solution.
First note that the system is homogeneous and hence it is consistent. Thus if the system has a nontrivial […]

Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix
Consider the following system of linear equations
\begin{align*}
2x+3y+z&=-1\\
3x+3y+z&=1\\
2x+4y+z&=-2.
\end{align*}
(a) Find the coefficient matrix $A$ for this system.
(b) Find the inverse matrix of the coefficient matrix found in (a)
(c) Solve the system using […]

Find Values of $a$ so that Augmented Matrix Represents a Consistent System
Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations.
\[A= \left[\begin{array}{rrr|r}
1 & 2 & 3 & 4 \\
2 &-1 & -2 & a^2 \\
-1 & -7 & -11 & a
\end{array} \right],\]
where $a$ is a real number. Determine all the […]

Determine Trigonometric Functions with Given Conditions
(a) Find a function
\[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\]
such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants.
(b) Find real numbers $a, b, c$ such that the function
\[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 […]

Solving a System of Linear Equations Using Gaussian Elimination
Solve the following system of linear equations using Gaussian elimination.
\begin{align*}
x+2y+3z &=4 \\
5x+6y+7z &=8\\
9x+10y+11z &=12
\end{align*}
Elementary row operations
The three elementary row operations on a matrix are defined as […]

Find a Basis For the Null Space of a Given $2\times 3$ Matrix
Let
\[A=\begin{bmatrix}
1 & 1 & 0 \\
1 &1 &0
\end{bmatrix}\]
be a matrix.
Find a basis of the null space of the matrix $A$.
(Remark: a null space is also called a kernel.)
Solution.
The null space $\calN(A)$ of the matrix $A$ is by […]

Determine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of Systems
Determine whether the following augmented matrices are in reduced row echelon form, and calculate the solution sets of their associated systems of linear equations.
(a) $\left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 6 \end{array} \right]$.
(b) […]

Write a Vector as a Linear Combination of Three Vectors
Write the vector $\begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}$ as a linear combination of the vectors
\[\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \, \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix} , \, \begin{bmatrix} 2 \\ 0 \\ 4 \end{bmatrix}.\]
Solution.
We want to find […]