Find the inverse of each unit triangular matrix 5 1 2 1 0 1

Find the inverse of each unit triangular matrix. 5. [1 2 -1 0 1 1 0 0 1] 6. [1 0 0 2 1 0 -3 2 1] Explain why the inverse of a unit upper (lower) triangular matrix is unit upper (lower) triangular.

Solution

5). first we have to find the determinant of the matrix

det = 1(1*1 - 0*1) - 2(0*1 - 0*1) +1(0*0 - 0*1)

= 1(1) -2(0) +1(0)

= 1 +0 +0

det = 1

now find adj of the matrix

M11= [ 1 1]

[0 1] { this we get from not taking the 1st row and column of the matix)

M11 = (1*1 - 0*1) =1

similerly find all other elements

M12 =0

M13=0

M21=1

M22=1

M23=0

M31=1

M32=1

M33=1

now inverse matrix = 1/det of matrix ( Adj of matrix)

inverse = [1 0 0]

[1 1 0]

[1 1 1]

in the inverse of upper(lowe) trinagular matrix is a upper(lower triangular since in adj of matrix , same will reflect