SATHEE: Matrices and Determinants 4 Question 25

Matrices and Determinants 4 Question 25

26. $A = [\begin{matrix} a & 0 & 1 \\ 1 & c & b \\ 1 & d & b \end{matrix}], B = [\begin{matrix} a & 1 & 1 \\ 0 & d & c \\ f & g & h \end{matrix}],$

$U = [\begin{matrix} f \\ g \\ h \end{matrix}],$ $V = | \begin{aligned} a^{2} \\ 0 \\ 0 \end{aligned} |$

If there is a vector matrix $X$ , such that $A X = U$ has infinitely many solutions, then prove that $B X = V$ cannot have a unique solution. If $a f d \neq 0$ . Then, prove that $B X = V$ has no solution.

$(2004, 4 M)$

Show Answer

Answer:

Correct Answer: 26. $- \sqrt{2} \leq λ \leq \sqrt{2}, α = n π, n π + \frac{π}{4}$

Solution:

Since, $A X = U$ has infinitely many solutions.

$\begin{aligned} \Rightarrow | A | = 0 \Rightarrow [\begin{array}{c} a & 0 & 1 \\ 1 & c & b \\ 1 & d & b \end{array}] & = 0 \\ \Rightarrow a (b c - b d) + 1 (d - c) & = 0 \Rightarrow (d - c) (a b - 1) = 0 \\ ∴ a b & = 1 or d = c \end{aligned}$

$\begin{aligned} Again, | A_{3} | = [\begin{array}{c} a & 0 & f \\ 1 & c & g \\ 1 & d & h \end{array}] = 0 \Rightarrow g = h \\ \Rightarrow | A_{2} | = [\begin{array}{c} a & f & 1 \\ 1 & g & b \\ 1 & h & b \end{array}] = 0 \Rightarrow g = h \\ and | A_{1} | = [\begin{array}{c} f & 0 & 1 \\ g & c & b \\ h & d & b \end{array}] \Rightarrow 0 \Rightarrow g = h \\ ∴ g = h, c = d and a b = 1 \\ Now, B X = V \\ | B | = [\begin{array}{c} a & 1 & 1 \\ 0 & d & c \\ f & g & h \end{array}] = 0 \end{aligned}$

[since, $C_{2}$ and $C_{3}$ are equal]

$∴ B X = V has no solution.$

$| B_{1} | = [\begin{matrix} a^{2} & 1 & 1 \\ 0 & d & c \\ 0 & g & h \end{matrix}] = 0$

$| B_{2} | = [\begin{matrix} a & a^{2} & 1 \\ 0 & 0 & c \\ f & 0 & h \end{matrix}] = a^{2} c f = a^{2} d f$

[since, $c = d$ and $g = h$ ]

$Since, a d f \neq 0 \Rightarrow | B_{2} | \neq 0$

$B X = V$ has no solution.

Matrices and Determinants 4 Question 25

26. $A = [\begin{matrix} a & 0 & 1 \\ 1 & c & b \\ 1 & d & b \end{matrix}], B = [\begin{matrix} a & 1 & 1 \\ 0 & d & c \\ f & g & h \end{matrix}],$

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Matrices and Determinants 4 Question 25

26. A=[a011cb1db],B=[a110dcfgh],

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

26. $A = [\begin{matrix} a & 0 & 1 \\ 1 & c & b \\ 1 & d & b \end{matrix}], B = [\begin{matrix} a & 1 & 1 \\ 0 & d & c \\ f & g & h \end{matrix}],$