The Identity Matrix and Diagonal Matrices

Practice Questions

Linear Algebra › The Identity Matrix and Diagonal Matrices

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By definition, a square matrix that is similar to a diagonal matrix is

True or false, if any of the main diagonal entries of a diagonal matrix is , then that matrix is not invertible.

Which of the following is true concerning the $n \times n$ identity matrix ?

Which of the following is true concerning diagonal matrices?

What is the minimum number of elementary row operations required to transform the identity matrix into its reduced row echelon form?

True or false:

$\begin{bmatrix} 1& 0&0 &0 \ \ \frac{4}{5}& 1 & 0 &0 \ \ - \frac{2}{5}& \frac{2}{5} & 1& 0\ \\frac{4}{5}& -\frac{6}{5} & -\frac{8}{5} & 1 \end{bmatrix}$

is an example of a diagonal matrix.

What is the name for a matrix obtained by performing a single elementary row operation on the identity matrix?

True or false:

$\begin{bmatrix} -0.1& 0 &0 &0 &0 &0 \ 0&0.2 &0 &0 &0 &0 \ 0& 0& -0.3&0 &0 &0 \ 0& 0& 0& 0.4& 0& 0\ 0& 0& 0& 0& -0.5& 0\ 0 &0 &0 &0 &0 & 0.6 \end{bmatrix}$

is an example of a diagonal matrix.

True or false:

$\begin{bmatrix} 2&0 & 0& 0 & 0\ 0&-7 &0 &0 &0 \ 0& 0& 5&0 & 0 \ 0& 0& 0& -4 & 0\ 0& 0& 0& 0 &0 \end{bmatrix}$

is an example of a diagonal matrix.

True or False, the $n \times n$ identity matrix has distinct (different) eigenvalues.

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