Question

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

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

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

Explanation:

Analyze the dimensions of the matrix

The given matrix has 5 rows and 5 columns:

$$ A =
LATEXBLOCK0
$$

Since the number of rows equals the number of columns, it is a square matrix.

Evaluate zero and identity properties

A zero matrix must have all entries equal to 0. Since the diagonal entries are 5, it is not a zero matrix.
An identity matrix must have 1s on the main diagonal and 0s elsewhere. Since the diagonal entries are 5, it is not the identity matrix.

Evaluate symmetry

A matrix is symmetric if it equals its transpose, \(A = A^T\).
For all \(i
eq j\), the entries \(a_{ij} = 0\), which satisfies \(a_{ij} = a_{ji}\). Thus, it is symmetric.

Evaluate diagonal and triangular properties

A diagonal matrix has all non-diagonal entries equal to 0. Since \(a_{ij} = 0\) for all \(i
eq j\), it is a diagonal matrix.
An upper triangular matrix has all entries below the main diagonal equal to 0. This is true for \(A\).
A lower triangular matrix has all entries above the main diagonal equal to 0. This is also true for \(A\).

Answer:

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