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DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS4 Marks
Question
Show that the following system of linear equations is consistent and also find solutions:
5x +3y + 7z = 4
3x + 26y + 2z = 9
7x + 2y + 10z = 5
✓

Answer

This can be written as:

$\begin{bmatrix}5&3&7\\ 3&26&2\\ 7&2&10\end{bmatrix}\begin{bmatrix}\text{X}\\ \text{y}\\ \text{z}\end{bmatrix}=\begin{bmatrix}4\\ 9\\ 5\end{bmatrix}$

Or $\text{A}\text{X}=\text{B}$

$\text{|A|}=5{(256)}-3{(16)}+7{(6-182)}$

$=0$

So, A is singular. Thus, the given system is either inconsistent or it is consistent with infinitely many solutions according as

$\text{(Adj A)}\times\text{B}\neq0$ or $\text{(Adj A)}\times\text{B}=0$

Let Cij be the co-factors of aij in A

$\text{C}_{11}=256\\ \text{C}_{21}=-16\\ \text{C}_{31}=-176$

$\text{C}_{12}=-16\\ \text{C}_{22}=1\\ \text{C}_{32}=11$

$\text{C}_{13}=-176\\ \text{C}_{23}=11\\ \text{C}_{33}=121$

$\text{adj A}=\begin{bmatrix}256&-16&-176\\ -16&1&11\\ -176&11&121\end{bmatrix}^\text{T}=\begin{bmatrix}256&-16&-176\\ -16&1&11\\ -176&11&121\end{bmatrix}$

$\text{adj}\text{A}\times\text{B}=\begin{bmatrix}256&-16&-176\\ -16&1&11\\ -176&11&121\end{bmatrix}\begin{bmatrix}4\\ 9\\ 5\end{bmatrix}=\begin{bmatrix}0\\ 0\\ 0\end{bmatrix}$

Thus, AX = B has infinite many solutions.

Now, let z = k

Then, 5x + 3y = 4 - 7k

3x + 26y = 9 - 2k

Which can be written as:

$\begin{bmatrix}5&3\\ 3&26\end{bmatrix}\begin{bmatrix}\text{x}\\ \text{y}\end{bmatrix}=\begin{bmatrix}4-7\text{k}\\ 9-2\text{k}\end{bmatrix}$

Or $\text{AX = B}$

$\text{|A|}=2$

$\text{adj A}=\begin{bmatrix}26&-3\\ -3&5\end{bmatrix}$

Now, $\text{x}=\text{A}^{-1}\text{B}=\frac{1}{\text{|A|}}\times\text{adj A}\times\text{B}$

$=\frac{1}{121}\begin{bmatrix}26&-3\\ -3&5\end{bmatrix}\begin{bmatrix}4-7\text{k}\\ 9-2\text{k}\end{bmatrix}$

$=\frac{1}{121}\begin{bmatrix}77-176\text{k}\\ 11\text{k}+33\end{bmatrix}$

$\begin{bmatrix}\text{x}\\ \text{y}\end{bmatrix}=\begin{bmatrix}\frac{7-16\text{k}}{11}\\ \frac{\text{k}+3}{11}\end{bmatrix}$

There values of x, y, z satisfies the third eq.

Hence, $\text{x}=\frac{7-16\text{k}}{11},\text{y}=\frac{\text{k}+3}{11},\text{z}=\text{k}$

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