From this, we see that the matrix $A$ is nonsingular if and only if the $(3, 3)$-entry $4a^2+a+1$ is not zero.
By the quadratic formula, we see that
are solutions of $4a^2+a+1=0$.
Note that these are not real numbers. Thus, for any real number $a$, we have $4a^2+a+1\neq 0$.
Hence, we can divide the third row by this number, and eventually we can reduce it to the identity matrix.
So the rank of $A$ is $3$, and $A$ is nonsingular for any real number $a$.
Determine whether the Given 3 by 3 Matrices are Nonsingular
Determine whether the following matrices are nonsingular or not.
1 & 0 & 1 \\
2 &1 &2 \\
1 & 0 & -1
2 & 1 & 2 \\
1 &0 &1 \\
4 & 1 & 4
Recall that […]
Find the Rank of a Matrix with a Parameter
Find the rank of the following real matrix.
a & 1 & 2 \\
1 &1 &1 \\
-1 & 1 & 1-a
where $a$ is a real number.
(Kyoto University, Linear Algebra Exam)
The rank is the number of nonzero rows of a […]
If Two Matrices Have the Same Rank, Are They Row-Equivalent?
If $A, B$ have the same rank, can we conclude that they are row-equivalent?
If so, then prove it. If not, then provide a counterexample.
Having the same rank does not mean they are row-equivalent.
For a simple counterexample, consider $A = […]
Find Values of $a$ so that the Matrix is Nonsingular
Let $A$ be the following $3 \times 3$ matrix.
1 & 1 & -1 \\
0 &1 &2 \\
1 & 1 & a
Determine the values of $a$ so that the matrix $A$ is nonsingular.
We use the fact that a matrix is nonsingular if and only if […]