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

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.

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 #2 : 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 #6 : The Identity Matrix And Diagonal Matrices

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 #3 : The Identity Matrix And Diagonal Matrices

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

**Possible Answers:**

False

True

**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 #8 : 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:**

None of the other answers

An elementary matrix

An elementary row matrix

A transition matrix

An inverse 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 #9 : The Identity Matrix And Diagonal Matrices

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

**Possible Answers:**

the identity matrix

None of the given answers

idempotent

diagonalizable

symmetric

**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.

is not diagonalizable.

has nullity .

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.

### All Linear Algebra Resources

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