Engineering Mat...

The solution to the system of equations \(\left[ {\begin{array}{*{20}{c}} 2&5\\ { - 4}&3 \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} x\\ y \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} 2\\ { - 30} \end{array}} \right\}\) is

What is the value of p if the solution of the given linear equations has infinitely many solutions

10a + 3b + 2c = 0

6a + 5b + c = 0

4a + pb + 5c = 0

In the above equations a, b, c are variable and p is contant.

Matrix for which LU decomposition is not possible?

Consider the system of equations (x + 3) p + 12q - 6x = 0, xp + (x - 5)q = 5x + 8.

If the system of equations has infinitely many solutions then what is the value of x?

Consider the below given homogeneous linear equation

2x – 4z + 6y + 2= 0

6z + 10y + 16 = 0

– 10z + 2x – 4y = 14

Consider a matrix M \(= \left[ {\begin{array}{*{20}{c}} 7&11\\ 5&9 \end{array}} \right]\;\)if M is going through LU decomposition and Matrix L diagonal elements are 1. What is the product of the main diagonal elements of Matrix U?

Among the given two sets which is/are consistent:   

\(S1 = \{ 3x + ay + 4z = 0,\;\;bx + 2y + z = \;0,\;\;5x + 7z + 9z = 0\)

\(S2 = \;\;\left\{ {2x + 6y = \; - 11,\;\;6x + 20y - \;6z = \; - 3,\;\;6y - 18z\; = \; - 1} \right\}\)

The number of purely real elements in a lower triangular representation of the given 3 × 3 matrix obtained through the given decomposition is _____

\(\left[ {\begin{array}{*{20}{c}} 2&3&3\\ 3&2&1\\ 3&1&7 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&0&0\\ {{a_{12}}}&{{a_{22}}}&0\\ {{a_{13}}}&{{a_{23}}}&{{a_{33}}} \end{array}} \right]{\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&0&0\\ {{a_{12}}}&{{a_{22}}}&0\\ {{a_{13}}}&{{a_{23}}}&{{a_{33}}} \end{array}} \right]^T}\)

If the coefficient matrix is \({C_{n \times n}}\) and its corresponding rank is c also the augmented matrix is \({A_{n \times \left( {n + 1} \right)}}\) and its corresponding rank is a, then how many statements given below are incorrect?

I. If c ≠ a, the equations are inconsistent with infinite number of solutions.

II. If c = a < n, the equations are inconsistent

The simultaneous equations

2x + ay + z = 20 

x + 3y + 4z = b

x + 2y + 3z = c

has unique solution then what is the value of a, b and c respectively?