Find the determinant of the nxn matrix A with 5's on the diagonal, l's above the diagonal, and O's below the diagonal. det (A) =

Mathematics

1 answer:

Vilka [71]3 years ago

6 0

Answer:

n times 5

Step-by-step explanation:

A matrix Anxn of this way is called an upper triangular matrix. It can be proved that the determinant of this kind of matrix is

$a_{11}+a_{22}+...+a_{nn}$

In this case, it would be 5+5+...+5 (n times) = n times 5

We are going to develop each determinant by the first column taking as pivot points the elements of the diagonal

$det\left[\begin{array}{cccc}5&a_{12}&a_{13}...&a_{1n}\\0&5&a_{23}...&a_{2n}\\...&...&...&...\\0&0&0&5\end{array}\right] =5+det\left[\begin{array}{ccc}5&a_{23}...&a_{2n}\\0&5&a_{3n}\\...&...&...\\0&0&5\end{array}\right]=5+5+...+det\left[\begin{array}{cc}5&a_{n-1,n}\\0&5\end{array}\right]=5+5+...+5+5\;(n\;times)$

You might be interested in

A committee consisting of 4 faculty members and 5 students is to be formed. Every committee position has the same duties and vot

Zina [86]

<u>Answer: </u>

The number of ways to form the committee is formed = 637065

<u>Solution: </u>

The number of combinations of n things such that m things taken together are given by:-

$C_{(\boldsymbol{n}, \boldsymbol{m})}=\frac{n !}{m !(\boldsymbol{n}-\boldsymbol{m}) !}$

Given Data:

A committee consisting of 4 faculty members and 5 students is to be formed. There are 12 faculty members and 13 students eligible to serve on the committee.

Then, the number of ways to form the committee is formed given by :-

$\begin{array}{l}{12 C_{4} \times 13 C_{5}} \\\\ {=\frac{12 !}{4 !(12-4) !} \times \frac{13 !}{5 !(13-5) !}} \\\\ {=\frac{12 !}{4 ! 8 !} \times \frac{13 !}{5 ! 8 !}} \\\\ {\frac{12 \times 11 \times 10 \times 9 \times 8 !}{27 \times 8 !} \times \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8}{120 \times 8 !}} \\\\ {=637065}\end{array}$