The Identity Matrix and Diagonal Matrices - Linear Algebra

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Question

Which of the following matrices is a scalar multiple of the identity matrix?

$\begin{pmatrix} 1&2 \ 3&4 \end{pmatrix}$ , $\begin{pmatrix} e&0 \ 0&e \end{pmatrix}$ , $\begin{pmatrix} 0& 2\ 2&0 \end{pmatrix}$

Answer

The x identity matrix is

$\begin{pmatrix} 1&0 \ 0&1 \end{pmatrix}$

For this problem we see that

$\begin{pmatrix} e&0 \ 0&e \end{pmatrix}=\begin{pmatrix} e1&e0 \ e0&e1 \end{pmatrix}=e\begin{pmatrix} 1&0 \ 0&1 \end{pmatrix}$

And so

$\begin{pmatrix} e&0 \ 0&e \end{pmatrix}$ is a scalar multiple of the identity matrix.

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Question

Which of the following is true concerning diagonal matrices?

Answer

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

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Question

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

Answer

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 .

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Question

If

$A = \begin{bmatrix} 2& 0& 0& 0& 0\ 0& 3& 0& 0& 0\ 0& 0& 4& 0& 0\ 0& 0& 0& 2& 0\ 0& 0& 0& 0& 2 \end{bmatrix}$

Find .

Answer

Since is a diagonal matrix, we can find it's powers more easily by raising the numbers inside it to the power in question.

$A^7 = \begin{bmatrix} 2^7& 0& 0& 0& 0\ 0& 3^7& 0& 0& 0\ 0& 0& 4^7& 0& 0\ 0& 0& 0& 2^7& 0\ 0& 0& 0& 0& 2^7 \end{bmatrix} = \begin{bmatrix} 128& 0& 0& 0& 0\ 0& 2187& 0& 0& 0\ 0& 0& 16384& 0& 0\ 0& 0& 0& 128& 0\ 0& 0& 0& 0& 128 \end{bmatrix}$

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Question

True or false, the set of all $n \times n$ diagonal matrices forms a subspace of the vector space of all $n \times n$ matrices.

Answer

To see why it's true, we have to check the two axioms for a subspace.

1. Closure under vector addition: is the sum of two diagonal matrices another diagonal matrix? Yes it is, only the diagonal entries are going to change, if at all. Nonetheless, it's still a diagonal matrix since all the other entries in the matrix are .

2. Closure under scalar multiplication: is a scalar times a diagonal matrix another diagonal matrix? Yes it is. If you multiply any number to a diagonal matrix, only the diagonal entries will change. All the other entries will still be .

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Question

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

Answer

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.

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Question

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

Answer

We can find the eigenvalues of the identity matrix by finding all values of $\lambda$ such that $det(I_n\lambda -I_n) =0$ .

Hence we have

$det(I_n\lambda -I_n) = (\lambda-1)^n = 0$

So $\lambda =1$ is the only eigenvalue, regardless of the size of the identity matrix.

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Question

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

Answer

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

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Question

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

Answer

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 $A=PBP^{-1}$ .

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Question

The $n \times n$ identity matrix

Answer

An idempotent matrix is one such that . This is satisfied by the identity matrix since the identity matrix times itself is once again the identity matrix.

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Question

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

Answer

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).

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Question

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.

Answer

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.

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Question

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.

Answer

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.

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Question

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.

Answer

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.

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Question

$A = \begin{bmatrix}1 &0 &0 &0 &0 \ 0& 0.3 & 0& 0& 0\ 0&0 &0.5 & 0& 0\ 0& 0& 0& 0.7& 0\ 0& 0& 0& 0& 0.9 \end{bmatrix}$

Which of the following is equal to $A ^{-1}$ ?

Answer

is a diagonal matrix - its only nonzero entries are in its main diagonal, which comprises the elements in row , column , for .

The inverse $A^{-1}$ of a such a matrix can be found simply by replacing each element in the main diagonal with its reciprocal. Rewriting the elements in the diagonal matrix as fractions,

$A = \begin{bmatrix}1 &0 &0 &0 &0 \ 0& \frac{3}{10} & 0& 0& 0\ 0&0 & \frac{5}{10} & 0& 0\ 0& 0& 0& \frac{7}{10}& 0\ 0& 0& 0&0& \frac{9}{10} \end{bmatrix}$ ,

or

$A = \begin{bmatrix}1 &0 &0 &0 &0 \ 0& \frac{3}{10} & 0& 0& 0\ 0&0 & \frac{1}{2} & 0& 0\ 0& 0& 0& \frac{7}{10}& 0\ 0& 0& 0&0& \frac{9}{10} \end{bmatrix}$ .

Replace each diagonal element with its reciprocal to find $A^{-1}$ :

$A^{-1} = \begin{bmatrix}1 &0 &0 &0 &0 \ 0& \frac{10}{3} & 0& 0& 0\ 0&0 &2 & 0& 0\ 0& 0& 0& \frac{10}{7}& 0\ 0& 0& 0&0& \frac{10}{9} \end{bmatrix}$

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Question

$A = \begin{bmatrix} e & 1+ i & 1 \ 0 & \pi & 1 - i \ 0 & 0 & i \end{bmatrix}$

True or false: is an example of a diagonal matrix.

Answer

A diagonal matrix has only zeroes as entries off of its main (upper-left to lower-right) diagonal. has three nonzero entries off this diagonal (Row 1, column 2, for example), so it is not a diagonal matrix.

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Question

$A = \begin{bmatrix} a& 1 & -2 \ -3 & a& 1 \ 1 & 2& a \end{bmatrix}$

is a strictly diagonally dominant matrix for what range of values of ?

Answer

An $n \times n$ matrix is strictly diagonally dominant if, for each $j \in 1, 2, ...n$ , it holds that on Row , the absolute value of the element in Column is greater than the sum of the absolute values of the other elements in that row. Therefore, for to be strictly diagonally dominant, the following must hold:

For Row 1:

or

For Row 2:

or

For Row 3: ,

or

For all three conditions to hold, it is necessary that . This is the correct choice.

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Question

is a diagonal matrix such that $A^{100} = 2 I$ , where refers to the $2 \times 2$ identity.

can be one of how many matrices?

Answer

is a diagonal matrix, and its dimensions are $2 \times 2$ , so, for some complex and (the problem did not specify that the entries were real),

$A = \begin{bmatrix} a & 0 \ 0 & b \end{bmatrix}$ .

To raise a diagonal matrix to a power, simply raise each nonzero element to that power. It holds that

$A^{100} = \begin{bmatrix} a ^{100} & 0 \ 0 & b^{100} \end{bmatrix} = \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix}$ ,

and, consequently, $a^{100} = b^{100} = 2$ .

Equivalently, both and must be a one-hundredth root of 2, of which there are 100. Therefore, the number of possible matrices for is $100 \times 100 = 10,000$ .

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Question

What is the trace of the $2 \times 3$ identity matrix?

Answer

This question does not make sense since there is no such thing as the $2 \times 3$ identity matrix, and it is not possible to take the trace of a matrix that is not square. This question is mostly meant to test your ability to think critically when reading certain mathematics problems.

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Question

True or false; all diagonal matrices are diagonalizable.

Answer

A matrix is diagonalizable if it can be written as $A = P^{-1}DA$ , where is any invertible matrix, and is any diagonal matrix. If is already a diagonal matrix, we can of course write it as $I^{-1}AI$ . Hence any diagonal matrix is diagonalizable.

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