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Official PYQ · 2017Set 2 · Q22Linear Algebra: Matrices, Determinants, System of Linear EquationsNAT1 mark

GATE CS 2017 Set 2 Q22 — Linear Algebra: Matrices, Determinants, System of Linear Equations

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Let P=[11−12−343−23]P = \begin{bmatrix}1 & 1 & -1 \\2 & -3 & 4 \\3 & -2 & 3\end{bmatrix}  and Q=[−1−2−161265105]Q = \begin{bmatrix}-1 & -2 &-1 \\6 & 12 & 6 \\5 & 10 & 5\end{bmatrix} be two matrices.

Then the rank of P+QP+Q is ___________ .
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Answer: 2

First, we compute the sum of the two matrices PP and QQ:

P+Q=[11−12−343−23]+[−1−2−161265105]=[0−1−28910888]P + Q = \begin{bmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{bmatrix} + \begin{bmatrix} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{bmatrix} = \begin{bmatrix} 0 & -1 & -2 \\ 8 & 9 & 10 \\ 8 & 8 & 8 \end{bmatrix}

To find the rank of P+QP+Q, we perform elementary row operations to reduce the matrix to row echelon form:

  1. Perform R3←R3−R2R_3 \leftarrow R_3 - R_2: [0−1−289100−1−2]\begin{bmatrix} 0 & -1 & -2 \\ 8 & 9 & 10 \\ 0 & -1 & -2 \end{bmatrix}
  2. Perform R3←R3−R1R_3 \leftarrow R_3 - R_1: [0−1−28910000]\begin{bmatrix} 0 & -1 & -2 \\ 8 & 9 & 10 \\ 0 & 0 & 0 \end{bmatrix}

The reduced matrix has 2 non-zero rows, which are linearly independent. Thus, the rank of P+QP+Q is 2.

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