Linear Algebra - The Identity Matrix and Diagonal Matrices

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

diagonalizable

idempotent

symmetric

the identity matrix

None of the given answers

Explanation

Another way to state this definition is that a square matrix is said to diagonalizable if and only if there exists some invertible matrix and diagonal matrix such that $A=PBP^{-1}$ .

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

True

False

Explanation

Probably the simplest way to see this is true is to take the determinant of the diagonal matrix. We can take the determinant of a diagonal matrix by simply multiplying all of the entries along its main diagonal. Since one of these entries is , then the determinant is , and hence the matrix is not invertible.

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

All of the other answers are true.

Explanation

is the trace operation. It means to add up the entries along the main diagonal of the matrix. Since has ones along its main diagonal, the trace of is .

Which of the following is true concerning diagonal matrices?

The product of two diagonal matrices (in either order) is always another diagonal matrix.

The determinant of any diagonal matrix is .

The zero matrix (of any size) is not a diagonal matrix.

The trace of any diagonal matrix is equal to its determinant.

All of the other answers are false.

Explanation

You can verify this directly by proving it, or by multiplying a few examples on your calculator.

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

Explanation

There is no need to perform any elementary row operations on the identity matrix; it is already in its reduced row echelon form. (There is a leading one in each row, and each column).

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.

False

True

Explanation

A matrix is diagonal if and only if $a_{m, n} = 0$ - that is, the element in column , row is equal to zero - for all . The given matrix violates this condition, since $a_{2,1}$ and five other elements are equal to nonzero numbers.

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

An elementary matrix

A transition matrix

An inverse matrix

An elementary row matrix

None of the other answers

Explanation

This is the correct term. Elementary matrices themselves can be used in place of elementary row operations when row reducing other matrices when convenient.

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

False

Explanation

A matrix is diagonal if and only if $a_{m, n} = 0$ - that is, the element in column , row is equal to zero - for all . The given matrix satisfies this condition, since its only nonzero elements are the first element in Column 1, the second element in Column 2, and so forth.

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

False

Explanation

A matrix is diagonal if and only if $a_{m, n} = 0$ - that is, the element in column , row is equal to zero - for all . The given matrix fits this criterion.

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