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Linear Algebra : Operations and Properties

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Example Questions

Example Question #1 : The Identity Matrix And Diagonal Matrices

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

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

Possible Answers:

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

$\begin{pmatrix} 0& 2\\ 2&0 \end{pmatrix}$

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

Correct answer:

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

Explanation:

The x identity matrix is

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

For this problem we see that

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

And so

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

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Example Question #1 : Operations And Properties

Which of the following is true concerning diagonal matrices?

Possible Answers:

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

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

The determinant of any diagonal matrix is .

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

All of the other answers are false.

Correct answer:

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

Explanation:

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

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Example Question #1 : The Identity Matrix And Diagonal Matrices

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

Possible Answers:

I_n^n = nI_n

Tr(I_n) = n

|I_n| = n

All of the other answers are true.

Correct answer:

Tr(I_n) = n

Explanation:

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

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Example Question #1 : Operations And Properties

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

Find A^7 .

Possible Answers:

None of the other answers

$\begin{bmatrix} 986& 0& 0& 0& 0\\ 0& 2953& 0& 0& 0\\ 0& 0& 1638& 0& 0\\ 0& 0& 0& 900& 0\\ 0& 0& 0& 0& 900 \end{bmatrix}$

$\begin{bmatrix} 64& 0& 0& 0& 0\\ 0& 433& 0& 0& 0\\ 0& 0& 10& 0& 0\\ 0& 0& 0& 98& 0\\ 0& 0& 0& 0& 65 \end{bmatrix}$

$\begin{bmatrix} 128& 0& 0& 0& 0\\ 0& 2187& 0& 0& 0\\ 0& 0& 16384& 0& 0\\ 0& 0& 0& 128& 0\\ 0& 0& 0& 0& 128 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 128& 0& 0& 0& 0\\ 0& 2187& 0& 0& 0\\ 0& 0& 16384& 0& 0\\ 0& 0& 0& 128& 0\\ 0& 0& 0& 0& 128 \end{bmatrix}$

Explanation:

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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Example Question #1 : Operations And Properties

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.

Possible Answers:

False

True

Correct answer:

True

Explanation:

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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Example Question #3 : Operations And Properties

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

Possible Answers:

False

True

Correct answer:

True

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.

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Example Question #1 : The Identity Matrix And Diagonal Matrices

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

Possible Answers:

True

False

Correct answer:

False

Explanation:

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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Example Question #5 : Operations And Properties

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

Possible Answers:

An elementary matrix

An inverse matrix

An elementary row matrix

None of the other answers

A transition matrix

Correct answer:

An elementary matrix

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.

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Example Question #6 : Operations And Properties

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

Possible Answers:

None of the given answers

idempotent

the identity matrix

diagonalizable

symmetric

Correct answer:

diagonalizable

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 $A=PBP^{-1}$ .

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Example Question #1 : The Identity Matrix And Diagonal Matrices

The $n \times n$ identity matrix

Possible Answers:

has distinct eigenvalues, regardless of size.

has rank .

is not diagonalizable.

is idempotent.

has nullity .

Correct answer:

is idempotent.

Explanation:

An idempotent matrix is one such that A^2 =A . This is satisfied by the identity matrix since the identity matrix times itself is once again the identity matrix.

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Linear Algebra : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #1 : The Identity Matrix And Diagonal Matrices

Example Question #1 : Operations And Properties

Example Question #1 : The Identity Matrix And Diagonal Matrices

Example Question #1 : Operations And Properties

Example Question #1 : Operations And Properties

Example Question #3 : Operations And Properties

Example Question #1 : The Identity Matrix And Diagonal Matrices

Example Question #5 : Operations And Properties

Example Question #6 : Operations And Properties

Example Question #1 : The Identity Matrix And Diagonal Matrices

All Linear Algebra Resources

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