Algebra - Linear

Question 1: Is multiplying $\begin{bmatrix}1&0\\0&1\end{bmatrix}$ a 2x2 matrix by $\begin{bmatrix}1\end{bmatrix}$ a 1x1 matrix possible?

No, the inner dimensions (2 and 1) don't match Yes, result is 1x1 Yes, result is 2x2 Yes, result is 2x1

Question 2: Is multiplying $\begin{bmatrix}1&0\\0&1\end{bmatrix}$ a 2x2 matrix by $\begin{bmatrix}1&0\end{bmatrix}$ a 1x2 matrix possible?

Yes, result is 2x1 No, the inner dimensions (2 and 1) don't match Yes, result is 1x2 Yes, result is 2x2

Question 3: Is multiplying $\begin{bmatrix}1&0\\0&1\end{bmatrix}$ a 2x2 matrix by $\begin{bmatrix}1\\0\end{bmatrix}$ a 2x1 matrix possible?

Yes, the result is a 2x1 matrix No, the dimensions don't match Yes, the result is a 2x2 matrix Yes, the result is a 1x2 matrix

Question 4: What is the result of multiplying the matrices $\begin{bmatrix}1&0\\0&1\end{bmatrix}$ by $\begin{bmatrix}1&0\\1&1\end{bmatrix}$?

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

Question 5: What is the result of multiplying the matrices $\begin{bmatrix}1&2\\1&3\end{bmatrix}$ by $\begin{bmatrix}1&1\\1&1\end{bmatrix}$?

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

Question 6: Is this the same result as multiplying $\begin{bmatrix}1&1\\1&1\end{bmatrix}$ by $\begin{bmatrix}1&2\\1&3\end{bmatrix}$?

No, matrix multiplication is not commutative No, because the matrices are different sizes Yes, matrix multiplication is always commutative Yes, because both matrices contain the same numbers

Question 7: When multiplying the matrix $\begin{bmatrix}1&1\\1&-1\end{bmatrix}$ by its inverse $\begin{bmatrix}0.5&0.5\\0.5&-0.5\end{bmatrix}$ what is the result?

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

Question 8: What is the result of the matrix multiplication $\begin{bmatrix}1&1\\2&2\end{bmatrix}$ squared?

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

Question 9: Is it possible to multiply the matrix $\begin{bmatrix}1&2&3\\1&2&3\end{bmatrix}$ by itself?

Yes, the result is 2x3 No, the inner dimensions (3 and 2) don't match Yes, the result is 3x3 Yes, the result is 2x2

Question 10: What could you change about the previous matrix (if anything) to make it multiplyable by itself?

Nothing, it already works Add a row of zeros Multiply it by its transpose instead (A x $A^T$ or $A^T$ x A) Remove the last column

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