Question

determine the type(s) of matrix. (check all that apply.)
\\\

$$\begin{bmatrix} 3 & -7 & 6 \\\\ 0 & 3 & 1 \\\\ 0 & 0 & 0 \\end{bmatrix}$$

\\

• identity
• lower triangular
• diagonal
• upper triangular
• symmetric
• square
• zero

Explanation:

Analyze matrix dimensions

Using the Matrix Classification knowledge point

$$ A =
LATEXBLOCK0
$$

The matrix has 3 rows and 3 columns, which means it is a square matrix.

Check triangular properties

Using the Triangular Matrix knowledge point

$$ a_{ij} = 0 \quad \text{for} \quad i > j $$

All entries below the main diagonal are zero:

$$ a_{21} = 0, \quad a_{31} = 0, \quad a_{32} = 0 $$

Thus, it is an upper triangular matrix.

Check symmetric properties

Using the Symmetric Matrix Properties knowledge point

$$ A^T =
LATEXBLOCK1
eq A $$

Since \(A^T
eq A\), the matrix is not symmetric.

Evaluate remaining types

Using the Matrix Classification knowledge point

• Not Identity: Diagonal entries are not all 1, and off-diagonals are not all 0.
• Not Lower triangular: Entries above the diagonal are non-zero.
• Not Diagonal: Off-diagonal entries like \(a_{12} = -7\) are non-zero.
• Not Zero: Contains non-zero entries.

Answer:

• Identity
• Lower triangular
• Diagonal
• Upper triangular (Correct answer)
• Symmetric
• Square (Correct answer)
• Zero