A square matrix of order n , A = [ aij ], is called upper triangular if aij = 0 for i > j, that is, all entries below the main diagonal are zero. (Lower triangular matrices are defined similarly and enjoy the same properties you are showing below). If M and N are two upper triangular matrices of order 3, show that (a) MN is upper triangular (b) |M| is the product of its diagonal elements. Note that (a) and (b) hold for any n .

B ) |M| is a product of its diagonal elements because the matrix of M is equal | M | = a1b2c3 from this | M | is a product of its diagonal elements for an value of n found in the

Step-by-step explanation:

[tex]\left[\begin{array}{ccc}a1&b1&c1\\0&b2&c2\\0&0&c3\end{array}\right][/tex] = M

M is a upper triangle with an order of three ( 3 )

N= [tex]\left[\begin{array}{ccc}d1&e1&f1\\0&e2&f2\\0&0&f3\end{array}\right][/tex]

N is also an upper triangle with an order of three ( 3 )

( A ) MN will be multiplying Matrix M and Matrix N

hence MN = [tex]\left[\begin{array}{ccc}a1d1&a1b1+b1e2&a1f1+b1f2+c1f3\\0&b2e2&b2f2+c2f3\\0&0&c3f3\end{array}\right][/tex]

from the product of MN it is clear that MN is also an upper triangle square matrix as well

( B ) |M| is a product of its diagonal elements because the matrix of M is equal | M | = a1b2c3 from this | M | is a product of its diagonal elements for an value of n found in the matrix