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Inverses Of Triangular Matrices

What Are Triangular Matrices ?

If a square matrix has all zeros below its main diagonal entries then it is called upper triangular matrix and if the square matrix has all zeros above the main diagonal entries then it is called lower triangular matrix. Therefore, a triangular matrix has either all zero entries above or below main diagonal.

Examples of Upper Triangular Matrix

A = \begin{bmatrix}a_{11} & a_{12}\\0 & a_{22}\end{bmatrix}B = \begin{bmatrix}a_{11} & a_{12}& a_{13}\\0 & a_{22}& a_{23}\\0 & 0 & a_{33}\end{bmatrix}C = \begin{bmatrix}2 & 1 & 9 & 3\\0 & -1 & 8 &-2\\0 & 0 & -7 & 5\\0& 0 & 0 & 4\end{bmatrix}

Examples of Lower Triangular Matrix

P = \begin{bmatrix}a_{11} & 0\\a_{12} & a_{22}\end{bmatrix}Q = \begin{bmatrix}a_{11} & 0& 0\\a_{21} & a_{22}& 0\\a_{31} & a_{32} & a_{33}\end{bmatrix}R = \begin{bmatrix}-1 & 0 & 0 & 0\\5 & 1 & 0 &0\\-3 & 1 & -7 & 0\\8& 2 & -6 & 2\end{bmatrix}

Rules Regarding Triangular Matrices

Here are some basic rules regarding the upper or lower triangular matrices.

  • If a square matrix A= [a_{ij}] is upper triangular matrix then a_{ij} = 0 and i < j.
  • If a square matrix A= [a_{ij}] is lower triangular matrix then a_{ij} = 0 and i > j.
  • If a square matrix A= [a_{ij}] is upper triangular matrix then i^{th} row has and starts with i - 1 zeros.
  • If a square matrix A= [a_{ij}] is lower triangular matrix then i^{th} row has and starts with j - 1 zeros.

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Inverses Of Triangular Matrices

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