which of these matrices are identity matrices? select all that apply.
which of these matrices are identity matrices? select all that apply.
Answer
Explanation:
Step1: Recall identity - matrix definition
An $n\times n$ identity matrix $I_n$ has 1s on the main - diagonal and 0s elsewhere.
Step2: Analyze the first matrix
The matrix $\begin{bmatrix}0&0&0&0\ - 1&-1&-1&-1\0&0&0&0\0&0&0&0\end{bmatrix}$ does not have 1s on the main - diagonal, so it is not an identity matrix.
Step3: Analyze the second matrix
The matrix $\begin{bmatrix}1&0\0&1\end{bmatrix}$ is a $2\times2$ matrix with 1s on the main - diagonal and 0s elsewhere, so it is an identity matrix.
Step4: Analyze the third matrix
The matrix $\begin{bmatrix}1&0&0&0\0&1&0&0\0&0&1&0\end{bmatrix}$ is not a square matrix. Identity matrices are square matrices, so it is not an identity matrix.
Step5: Analyze the fourth matrix
The matrix $\begin{bmatrix}1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1\end{bmatrix}$ is a $4\times4$ matrix with 1s on the main - diagonal and 0s elsewhere, so it is an identity matrix.
Answer:
The identity matrices are $\begin{bmatrix}1&0\0&1\end{bmatrix}$ and $\begin{bmatrix}1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1\end{bmatrix}$.