If a matrix is both upper triangular and lower triangular, what type of matrix i

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Question: If a matrix is both upper triangular and lower triangular, what type of matrix is it? (2020)

Options:

Correct Answer: Diagonal matrix

Exam Year: 2020

Solution:

A matrix that is both upper and lower triangular must have non-zero elements only on the diagonal, making it a diagonal matrix.

If a matrix is both upper triangular and lower triangular, what type of matrix is it? (2020)

If a matrix is both upper triangular and lower triangular, what type of matrix is it? (2020)

Step 1: Understand what an upper triangular matrix is. An upper triangular matrix has all its elements below the main diagonal equal to zero.
Step 2: Understand what a lower triangular matrix is. A lower triangular matrix has all its elements above the main diagonal equal to zero.
Step 3: If a matrix is both upper and lower triangular, it means that all elements below and above the main diagonal are zero.
Step 4: The only elements that can be non-zero are those on the main diagonal.
Step 5: Therefore, a matrix that is both upper and lower triangular must have non-zero elements only on the diagonal.
Step 6: This type of matrix is called a diagonal matrix.

Upper Triangular Matrix – A matrix where all the entries below the main diagonal are zero.
Lower Triangular Matrix – A matrix where all the entries above the main diagonal are zero.
Diagonal Matrix – A matrix where all off-diagonal entries are zero, and only diagonal entries may be non-zero.