Solution 1: Given matrix is.

Solution 1: Given matrix is
Let’s transform the matrix into upper triangular using elementary row operations.
, we get
We know that the determinant of upper triangular matrix is the product of the diagonal
entries. Therefore, the determinant of the required matrix is
Solution 2: Given matrix is
Now,
The characteristic polynomial is
Therefore, the characteristic polynomial is
Solution 3: Given that L be the linear transformation on R2 that reflects each point P
across the line
a): It is observe that vector is on the line this implies that
. Hence, is the eigenvector of L, where A is the matrix of the linear
transformation.
Now, for , the line is perpendicular to the line at the origin and if
, the line perpendicular to the line at the origin
From above, it is observed that in either case the vector is on the perpendicular line
So, by the reflection across the line , this vector is mapped to
This implies that
Hence, is the eigenvector of L.
b): Since, this implies that . So 1 is the eigenvalue.
Since, this implies that . So is the eigenvalue.
Solution 4: Given a linear transformation defined as