Triangular matrix (Upper and Lower triangular matrix) - Practice Questions & MCQ

Triangular matrix: A square matrix whose all elements above or below the principal diagonal are zero is called a Triangular matrix.

A triangular matrix is further classified into two types:

 Upper triangular matrix

 Lower triangular matrix

Upper triangular matrix: A square matrix whose all elements below the principal diagonal are zero is called an upper triangular matrix.

Or $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be upper triangular if $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}=0$ when $\mathrm{i}>\mathrm{j}$.
E.g.,

$
\left[\begin{array}{ccccc}
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
0 & a_{22} & a_{23} & a_{24} & a_{25} \\
0 & 0 & a_{33} & a_{34} & a_{35} \\
0 & 0 & 0 & a_{44} & a_{45} \\
0 & 0 & 0 & 0 & a_{55}
\end{array}\right]
$

Lower triangular matrix: A square matrix whose all elements above the principal diagonal are zero is called a lower triangular matrix.

Or $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be upper triangular if $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}=0$ when $\mathrm{i}<\mathrm{j}$. Eg.

$
\left[\begin{array}{ccccc}
a_{11} & 0 & 0 & 0 & 0 \\
a_{21} & a_{22} & 0 & 0 & 0 \\
a_{31} & a_{32} & a_{33} & 0 & 0 \\
a_{41} & a_{42} & a_{43} & a_{44} & 0 \\
a_{51} & a_{52} & a_{53} & a_{54} & a_{55}
\end{array}\right]
$