A value of b for which the rank of the matrix is $$A=11-1044-31b22299b3is$$ $$3$$, is

$$11-1044-31b22299b3R2$$→$$R2$$−$$4R1R3$$→$$R3$$−$$bR1R4$$→$$R4$$−$$9R1$$~$$11-10001102-b2+b200b+93R2$$↔$$R3$$~$$11-1002-b2+b2001100b+93R4$$→$$R4$$−$$3R3$$~$$11-1002-b2+b2001100b+60Rank$$ $$ofA=3$$⇒$$b+6=0$$ ⇒$$b=$$−$$6$$

Elementary row operations (Transformation)

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A value of $b$ for which the rank of the matrix is $A = [\begin{matrix} 1 & 1 & - 1 & 0 \\ 4 & 4 & - 3 & 1 \\ b & 2 & 2 & 2 \\ 9 & 9 & b & 3 \end{matrix}]$ is 3, is

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Solution

$[\begin{matrix} 1 & 1 & - 1 & 0 \\ 4 & 4 & - 3 & 1 \\ b & 2 & 2 & 2 \\ 9 & 9 & b & 3 \end{matrix}]$ $\begin{array}{l} R_{2} \to R_{2} - 4 R_{1} \\ R_{3} \to R_{3} - b R_{1} \\ R_{4} \to R_{4} - 9 R_{1} \end{array}$
$~ [\begin{matrix} 1 & 1 & - 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 2 - b & 2 + b & 2 \\ 0 & 0 & b + 9 & 3 \end{matrix}]$ $R_{2} \leftrightarrow R_{3}$
$~ [\begin{matrix} 1 & 1 & - 1 & 0 \\ 0 & 2 - b & 2 + b & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & b + 9 & 3 \end{matrix}]$ $R_{4} \to R_{4} - 3 R_{3}$
$~ [\begin{matrix} 1 & 1 & - 1 & 0 \\ 0 & 2 - b & 2 + b & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & b + 6 & 0 \end{matrix}]$
Rank of $A = 3 \Rightarrow b + 6 = 0$
$\Rightarrow b = - 6$

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Elementary row operations (Transformation)

Remember concepts with our Masterclasses.

A value of b for which the rank of the matrix is A=11-1044-31b22299b3is 3, is

11-1044-31b22299b3R2→R2−4R1R3→R3−bR1R4→R4−9R1~11-10001102-b2+b200b+93R2↔R3~11-1002-b2+b2001100b+93R4→R4−3R3~11-1002-b2+b2001100b+60Rank ofA=3⇒b+6=0 ⇒b=−6

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A value of $b$ for which the rank of the matrix is $A = [\begin{matrix} 1 & 1 & - 1 & 0 \\ 4 & 4 & - 3 & 1 \\ b & 2 & 2 & 2 \\ 9 & 9 & b & 3 \end{matrix}]$ is 3, is