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Linear Algebra : The Identity Matrix and Diagonal Matrices

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Example Questions

Example Question #21 : The Identity Matrix And Diagonal Matrices

Which of the following is an identity matrix?

Possible Answers:

All of these are valid identitiy matricies.

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

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

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

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

Correct answer:

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

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:

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

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 #22 : The Identity Matrix And Diagonal Matrices

Which of the following is a diagonal matrix?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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Example Question #23 : The Identity Matrix And Diagonal Matrices

Which of the following is a diagonal matrix?

Possible Answers:

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

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

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

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

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

Correct answer:

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

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:

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

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Linear Algebra : The Identity Matrix and Diagonal Matrices

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #21 : The Identity Matrix And Diagonal Matrices

Example Question #22 : The Identity Matrix And Diagonal Matrices

Example Question #23 : The Identity Matrix And Diagonal Matrices

All Linear Algebra Resources

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