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Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #1 : Operations And Properties

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

Possible Answers:

Tr(I_n) = n

|I_n| = n

All of the other answers are true.

I_n^n = nI_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 #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:

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

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

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

None of the other answers

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

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 #81 : Linear Algebra

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 : 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:

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 #4 : The Identity Matrix And Diagonal Matrices

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 #5 : 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 matrix

None of the other answers

A transition 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 #3 : The Identity Matrix And Diagonal Matrices

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

Possible Answers:

the identity matrix

diagonalizable

None of the given answers

symmetric

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 #7 : The Identity Matrix And Diagonal Matrices

The $n \times n$ identity matrix

Possible Answers:

has rank .

is idempotent.

has nullity .

has distinct eigenvalues, regardless of size.

is not diagonalizable.

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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Example Question #11 : The Identity Matrix And Diagonal Matrices

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

Possible Answers:

Correct answer:

Explanation:

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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Example Question #12 : The Identity Matrix And Diagonal Matrices

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.

Possible Answers:

True

False

Correct answer:

False

Explanation:

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 $m \ne n$ . The given matrix violates this condition, since $a_{2,1}$ and five other elements are equal to nonzero numbers.

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Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #1 : Operations And Properties

Example Question #1 : Operations And Properties

Example Question #81 : Linear Algebra

Example Question #2 : The Identity Matrix And Diagonal Matrices

Example Question #4 : The Identity Matrix And Diagonal Matrices

Example Question #5 : The Identity Matrix And Diagonal Matrices

Example Question #3 : The Identity Matrix And Diagonal Matrices

Example Question #7 : The Identity Matrix And Diagonal Matrices

Example Question #11 : The Identity Matrix And Diagonal Matrices

Example Question #12 : The Identity Matrix And Diagonal Matrices

All Linear Algebra Resources

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