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Linear Algebra : Operations and Properties

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Example Question #91 : Linear Algebra

Which of the following is an identity matrix?

Possible Answers:

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

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

All of these are valid identitiy matricies.

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

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

Correct answer:

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

Explanation:

An identity matrix is a square matrix in which all diagonal elements are 1 and all non-diagonal elements are 0. The only square matricies above are:

$\begin{bmatrix} 1& 0& 0\\ 0& 1& 0\\ 0&0 &1 \end{bmatrix} \textup{and}$ $\begin{bmatrix} 0& 0&1 \\ 0& 1 & 0\\ 1& 0&0 \end{bmatrix}$

The former of these two matricies is the only one fitting the critera of diagonal elements (diagonal meaning from top left to bottom right) being 1 and non-diagonal elements being 0.

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Example Question #91 : Linear Algebra

Which of the following is a diagonal matrix?

Possible Answers:

$\begin{bmatrix} 0& 1& 2\\ 3& 0& 4\\ 5&6 & 0 \end{bmatrix}$

$\begin{bmatrix} 1&0 &2 \\ 0& 3& 0\\ 4& 0& 5 \end{bmatrix}$

$\begin{bmatrix} 0& 0& 1\\ 0& 2& 0\\ 3&0 & 0 \end{bmatrix}\,$

$\begin{bmatrix} 1& 0& 0\\ 0& 2& 0\\ 0&0 & 3 \end{bmatrix}$

$\begin{bmatrix} 1& 2&0 &0 \\ 0& 0& 3& 4 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 1& 0& 0\\ 0& 2& 0\\ 0&0 & 3 \end{bmatrix}$

Explanation:

$\begin{bmatrix} 1& 0& 0\\ 0& 2& 0\\ 0&0 & 3 \end{bmatrix}$

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Example Question #21 : Operations And Properties

Which of the following is a diagonal matrix?

Possible Answers:

$\begin{bmatrix} 0&9 &0 &0 \\ 0 & 0& 3& 0\\ 0 & 0& 0& 1 \end{bmatrix}\:$

$\begin{bmatrix} 3&0 &0 &0 \\ 0 & 7& 0& 0\\ 0 & 0& 1& 9 \end{bmatrix}$

$\begin{bmatrix} 3&0 &0 &0 \\ 0 & 7& 5& 0\\ 0 & 0& 0& 6 \end{bmatrix}$

$\begin{bmatrix} 2&0 &0 &0 \\ 0 & 4& 0& 0\\ 0 & 0& 7& 0 \end{bmatrix}$

$\begin{bmatrix} 4&0 &0 &0 \\ 0 & 5& 0& 0\\ 0 & 0& 0& 2 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 2&0 &0 &0 \\ 0 & 4& 0& 0\\ 0 & 0& 7& 0 \end{bmatrix}$

Explanation:

A diagonal matrix is a matrix in which all elements except diagonal elements are zero. A diagonal element of a matrix M is the element M_ij for which i=j. Here, i refers to the number of columns and j is the number of rows. The matrix doesn't necessarily have to be a square matrix; as long as elements not in the form M_ii are zero, the matrix is diagnoal. The only matrix that fufills this definition is:

$\begin{bmatrix} 2&0 &0 &0 \\ 0 & 4& 0& 0\\ 0 & 0& 7& 0 \end{bmatrix}$

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Example Question #1 : The Transpose

Find the transpose of Matrix .

$A=\begin{bmatrix} 1& 4\\ 5& 8 \end{bmatrix}$

Possible Answers:

$A^T=\begin{bmatrix} 1& 5\\ 8& 4 \end{bmatrix}$

$A^T=\begin{bmatrix} 1& 4\\ 5& 8 \end{bmatrix}$

$A^T=\begin{bmatrix} 8& 5\\ 4& 1 \end{bmatrix}$

$A^T=\begin{bmatrix} 4& 5\\ 8& 1 \end{bmatrix}$

$A^T=\begin{bmatrix} 1& 5\\ 4& 8 \end{bmatrix}$

Correct answer:

$A^T=\begin{bmatrix} 1& 5\\ 4& 8 \end{bmatrix}$

Explanation:

To find the transpose, we need to make columns into rows.

$A^T=\begin{bmatrix} 1& 5\\ 4& 8 \end{bmatrix}$

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Example Question #2 : The Transpose

Transpose matrix A where,

$A=\begin{bmatrix} 5&4 &7 \\ 8&4 &2 \\ 0&9 &5 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 7&4 &5 \\ 2&4 &8 \\ 5&9 &0 \end{bmatrix}$

$\begin{bmatrix} 5&4 &7 \\ 0&9 &5 \\ 8&4 &2 \end{bmatrix}$

$\begin{bmatrix} 5&8 &0 \\ 4&4 &9 \\ 7&2 &5 \end{bmatrix}$

You cannot transpose a square matrix

$\begin{bmatrix} 4&4 &9 \\ 5&8 &0 \\ 7&2 &5 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 5&8 &0 \\ 4&4 &9 \\ 7&2 &5 \end{bmatrix}$

Explanation:

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: $\begin{bmatrix} 1& 2&3 \\ 4&5 &6 \\ 7&8 &9 \end{bmatrix}^{T} = \begin{bmatrix} 1&4 &7 \\ 2&5 &8 \\ 3&6 &9 \end{bmatrix}$ ie. column 1 become row 1, column 2 becomes row 2, etc.

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Example Question #3 : The Transpose

$\begin{bmatrix} 2&5 \\ 4& 5\\ 3&2 \\ 0&1 \end{bmatrix}^{T}=$

Possible Answers:

$\begin{bmatrix} 2&4 &3 &0 \\ 5&5 &2 &1 \end{bmatrix}$

Not possible with non-square matrices

$\begin{bmatrix} 5&2 \\ 5&4 \\ 2&3 \\ 1&0 \end{bmatrix}$

$\begin{bmatrix} 5&5 &2 &1 \\ 2&4 &3 &0 \end{bmatrix}$

$\begin{bmatrix} 2&5 &3 &1 \\ 5& 4&2 &0 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 2&4 &3 &0 \\ 5&5 &2 &1 \end{bmatrix}$

Explanation:

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Example Question #4 : The Transpose

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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Example Question #5 : The Transpose

$\begin{bmatrix} 2&10 &54 \\ -5& 6& 17\\ 25& 4& 0\\ 2& 12&-4 \end{bmatrix}^{T}=$

Possible Answers:

$\begin{bmatrix} 54&17 &0 &-4 \\ 10&6 &4 &12 \\ 2&-5 &25 &2 \end{bmatrix}$

$\begin{bmatrix} 54&10 &2 \\ 17&6 &-5 \\ 0&4 &25 \\ -4&12 &2 \end{bmatrix}$

$\begin{bmatrix} 6&25 &17 &54 \\ 0& -4& -5& 2\\ 4&10 &12 &2 \end{bmatrix}$

$\begin{bmatrix} 10& 6& 4&12 \\ 2&-5 &25 &2 \\ 54&17 &0 &-4 \end{bmatrix}$

$\begin{bmatrix} 2&-5 &25 &2 \\ 10&6 &4 &12 \\ 54&17 &0 &-4 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 2&-5 &25 &2 \\ 10&6 &4 &12 \\ 54&17 &0 &-4 \end{bmatrix}$

Explanation:

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Example Question #6 : The Transpose

$\begin{bmatrix} 7&8 \\ 0&2 \end{bmatrix}^{T}=$

Possible Answers:

Not Possible

$\begin{bmatrix} 8&7 \\ 2&0 \end{bmatrix}$

$\begin{bmatrix} 2&8 \\ 0&7 \end{bmatrix}$

$\begin{bmatrix} 7&0 \\ 8&2 \end{bmatrix}$

$\begin{bmatrix} 8& 2\\ 7& 0 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 7&0 \\ 8&2 \end{bmatrix}$

Explanation:

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Example Question #7 : The Transpose

$\begin{bmatrix} 5\\ -8\\ 10 \end{bmatrix}^{T}=$

Possible Answers:

$\begin{bmatrix} 5&-8 &10 \end{bmatrix}$

$\begin{bmatrix} 10\\ -8\\ 5 \end{bmatrix}$

$\begin{bmatrix} -5&8 &-10 \end{bmatrix}$

$\begin{bmatrix} -5\\8 \\-10 \end{bmatrix}$

$\begin{bmatrix} 10&-8 &5 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 5&-8 &10 \end{bmatrix}$

Explanation:

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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 #91 : Linear Algebra

Example Question #91 : Linear Algebra

Example Question #21 : Operations And Properties

Example Question #1 : The Transpose

Example Question #2 : The Transpose

Example Question #3 : The Transpose

Example Question #4 : The Transpose

Example Question #5 : The Transpose

Example Question #6 : The Transpose

Example Question #7 : The Transpose

All Linear Algebra Resources

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