Find the rank of the following matrices- A- 1 5 9- 4 8 12- 7 11 15 -

Minor of highest order 3 is #160; [ #160; 1 amp;5 amp;9; 4 amp;8 amp;12; 7 amp; 11 amp;15 ] = 0by R_1+R_3 2R_2 Minor of order 2 is ...

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Standard XII

Mathematics

Elementary Transformations

Find the rank...

Question

Find the rank of the following matrices:
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 5 & 9 4 & 8 & 12 7 & 11 & 15 \end{matrix} ⎤ ⎥ ⎦$ .

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Solution

Minor of highest order 3 is
$⎡ ⎢ ⎣ \begin{matrix} 1 & 5 & 9 4 & 8 & 12 7 & 11 & 15 \end{matrix} ⎤ ⎥ ⎦$ = 0
by $R_{1} + R_{3} - 2 R_{2}$
Minor of order 2 is $[\begin{matrix} 1 & 5 4 & 8 \end{matrix}]$ = 8-20 = -12 $\neq$ 0
Hence p(A) = 2
Note. Rank of unit matrix, $I_{3}$ is 3 as minor of highest order 3 is | $I_{3}$ | = 1 $\neq$ 0. Similarly rank $I_{n}$ = n.

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Find the rank of the following matrices:A=⎡⎢⎣159481271115⎤⎥⎦.

Minor of highest order 3 is ⎡⎢⎣159481271115⎤⎥⎦= 0byR1+R3−2R2Minor of order 2 is [1548]= 8-20 = -12 ≠ 0Hence p(A) = 2Note. Rank of unit matrix, I3 is 3 as minor of highest order 3 is |I3 | = 1 ≠ 0. Similarly rank In = n.

Find the rank of the following matrices:
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 5 & 9 4 & 8 & 12 7 & 11 & 15 \end{matrix} ⎤ ⎥ ⎦$ .