Find Values of $a$ so that the Matrix is Nonsingular

Problem 126

Let $A$ be the following $3 \times 3$ matrix.
\[A=\begin{bmatrix}
1 & 1 & -1 \\
0 &1 &2 \\
1 & 1 & a
\end{bmatrix}.\]
Determine the values of $a$ so that the matrix $A$ is nonsingular.

If $a+1=0$, then the last matrix is in reduced row echelon form.
Thus $A$ is not row equivalent to the identity matrix.

On the other hand, if $a+1 \neq 0$, then we can continue the reduction as follows.
\begin{align*}
\begin{bmatrix}
1 & 0 & -3 \\
0 &1 &2 \\
0 & 0 & a+1
\end{bmatrix}
\xrightarrow{\frac{1}{a+1} R_3}
\begin{bmatrix}
1 & 0 & -3 \\
0 &1 &2 \\
0 & 0 & 1
\end{bmatrix}
\xrightarrow[R_2-2R_3]{R_1+3R_3}
\begin{bmatrix}
1 & 0 & 0 \\
0 &1 &0 \\
0 & 0 & 1
\end{bmatrix}.
\end{align*}
Therefore $A$ is row equivalent to the identity matrix.

We conclude that the matrix $A$ is nonsingular for any values of $a$ except for $a=-1$.

Comment.

If you know how to compute the determinant of a $3 \times 3$ matrix, then you may also solve this using the fact that a matrix is nonsingular if and only if the determinant of it is nonzero.

Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.
(a) $A=\begin{bmatrix}
1 & 3 & -2 \\
2 &3 &0 \\
[…]

Possibilities For the Number of Solutions for a Linear System
Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.
(a) \[\left\{
\begin{array}{c}
ax+by=c \\
dx+ey=f,
\end{array}
\right.
\]
where $a,b,c, d$ […]

If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix
Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix.
Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.
Proof.
Because $A$ has rank $n$, we know that the $n \times n$ […]

Express a Vector as a Linear Combination of Other Vectors
Express the vector $\mathbf{b}=\begin{bmatrix}
2 \\
13 \\
6
\end{bmatrix}$ as a linear combination of the vectors
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
5 \\
-1
\end{bmatrix},
\mathbf{v}_2=
\begin{bmatrix}
1 \\
2 \\
1
[…]

Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix.
Let A be the matrix
\[\begin{bmatrix}
1 & -1 & 0 \\
0 &1 &-1 \\
0 & 0 & 1
\end{bmatrix}.\]
Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse.
(The Ohio State University Linear Algebra […]

Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix
Let \[A=\begin{bmatrix}
1 & 1 & 2 \\
2 &2 &4 \\
2 & 3 & 5
\end{bmatrix}.\]
(a) Find a matrix $B$ in reduced row echelon form such that $B$ is row equivalent to the matrix $A$.
(b) Find a basis for the null space of $A$.
(c) Find a basis for the range of $A$ that […]

Subspaces of Symmetric, Skew-Symmetric Matrices
Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.
(a) The set $S$ consisting of all $n\times n$ symmetric matrices.
(b) The set $T$ consisting of […]

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 […]

Let $S$ be the following subset of the 3-dimensional vector space $\R^3$. \[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1...