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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 : The Identity Matrix And Diagonal Matrices

Which of the following is true concerning diagonal matrices?

Possible Answers:

The determinant of any diagonal matrix is .

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

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

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

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

All of the other answers are true.

|I_n| = n

Tr(I_n) = n

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 #2 : The Identity Matrix And Diagonal Matrices

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

Find A^7 .

Possible Answers:

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

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

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

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

None of the other answers

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 #2 : 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:

True

False

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 #3 : Operations And Properties

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

Possible Answers:

False

True

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 #3 : The Identity Matrix And Diagonal Matrices

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

Possible Answers:

A transition matrix

None of the other answers

An elementary matrix

An elementary row matrix

An inverse 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 #4 : Operations And Properties

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

Possible Answers:

None of the given answers

diagonalizable

symmetric

the identity matrix

idempotent

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 #5 : Operations And Properties

The $n \times n$ identity matrix

Possible Answers:

has nullity .

has distinct eigenvalues, regardless of size.

is not diagonalizable.

has rank .

is idempotent.

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 : The Identity Matrix And Diagonal Matrices

Example Question #1 : The Identity Matrix And Diagonal Matrices

Example Question #2 : The Identity Matrix And Diagonal Matrices

Example Question #1 : Operations And Properties

Example Question #2 : Operations And Properties

Example Question #3 : Operations And Properties

Example Question #3 : The Identity Matrix And Diagonal Matrices

Example Question #4 : Operations And Properties

Example Question #5 : Operations And Properties

All Linear Algebra Resources

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