### All Linear Algebra Resources

## Example Questions

### Example Question #1 : The Identity Matrix And Diagonal Matrices

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

, ,

**Possible Answers:**

**Correct answer:**

The x identity matrix is

For this problem we see that

And so

is a scalar multiple of the identity matrix.

### Example Question #1 : The Identity Matrix And Diagonal Matrices

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 trace of any diagonal matrix is equal to its determinant.

All of the other answers are false.

The determinant of any diagonal matrix is .

**Correct answer:**

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

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

### Example Question #3 : The Identity Matrix And Diagonal Matrices

Which of the following is true concerning the identity matrix ?

**Possible Answers:**

All of the other answers are true.

**Correct 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 .

### Example Question #4 : The Identity Matrix And Diagonal Matrices

If

Find .

**Possible Answers:**

None of the other answers

**Correct 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.

### Example Question #1 : The Identity Matrix And Diagonal Matrices

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

**Possible Answers:**

False

True

**Correct answer:**

True

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 .

### Example Question #1 : 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

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.

### Example Question #1 : The Identity Matrix And Diagonal Matrices

True or False, the identity matrix has distinct (different) eigenvalues.

**Possible Answers:**

True

False

**Correct answer:**

False

We can find the eigenvalues of the identity matrix by finding all values of such that .

Hence we have

So is the only eigenvalue, regardless of the size of the identity matrix.

### Example Question #1 : 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:**

An elementary row matrix

A transition matrix

None of the other answers

An inverse matrix

An elementary matrix

**Correct answer:**

An elementary matrix

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

### Example Question #1 : The Identity Matrix And Diagonal Matrices

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

**Possible Answers:**

idempotent

the identity matrix

symmetric

diagonalizable

None of the given answers

**Correct answer:**

diagonalizable

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 .

### Example Question #10 : The Identity Matrix And Diagonal Matrices

The identity matrix

**Possible Answers:**

is idempotent.

has nullity .

is not diagonalizable.

has rank .

has distinct eigenvalues, regardless of size.

**Correct answer:**

is idempotent.

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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