Let
\[A=[A_1, A_2]=\begin{bmatrix}
3 & -2\\
7& 4
\end{bmatrix},\]
be the coefficient matrix of the system, where $A_1, A_2$ are column vectors of $A$.
Let $\mathbf{b}=\begin{bmatrix}
5 \\
-1
\end{bmatrix}$ be the constant term vector. Then the system can be written as
\[A\mathbf{x}=\mathbf{b},\]
where $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}$.

We form
\[B_1=[\mathbf{b}, A_2]=\begin{bmatrix}
5 & -2\\
-1& 4
\end{bmatrix}\]
and
\[B_2=[A_1, \mathbf{b}]=\begin{bmatrix}
3 & 5\\
7& -1
\end{bmatrix}.\]

Then Cramer’s rule gives the formula for solutions
\[x_1=\frac{\det(B_1)}{\det(A)} \text{ and } x_2=\frac{\det(B_2)}{\det(A)}. \tag{*}\]
Thus, it remains to compute the determinants.
We have
\begin{align*}
\det(A)=\begin{vmatrix}
3 & -2\\
7& 4
\end{vmatrix}=3\cdot 4 -(-2)\cdot 7 =26.
\end{align*}
Similarly, a calculation shows that
\[\det(B_1)=18 \text{ and } \det(B_2)=-38.\]

Therefore by Cramer’s rule (*), we obtain
\[x_1=\frac{18}{26}=\frac{9}{13} \text{ and } x_2=\frac{-38}{26}=-\frac{19}{13}.\]

Solving a System of Linear Equations By Using an Inverse Matrix
Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}
(a) Find the coefficient matrix and its inverse matrix.
(b) Using the inverse matrix, solve the system of linear equations.
(The Ohio […]

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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 […]

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 […]

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In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems.
Determine all possibilities for the solution set of the system of linear equations described below.
(a) A homogeneous system of $3$ […]

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Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
(1) \[S_1=\left \{\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\]
in […]

Quiz 11. Find Eigenvalues and Eigenvectors/ Properties of Determinants
(a) Find all the eigenvalues and eigenvectors of the matrix
\[A=\begin{bmatrix}
3 & -2\\
6& -4
\end{bmatrix}.\]
(b) Let
\[A=\begin{bmatrix}
1 & 0 & 3 \\
4 &5 &6 \\
7 & 0 & 9
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 0 & 0 \\
0 & 3 &0 […]

Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation
(a) Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 0 & 1 \\
1 &0 &0 \\
2 & 1 & 1
\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.
(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\]
where […]