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Quantitative Aptitude

Maxima and Minima

Find the matr...

Question

Find the matrix A such that
(i) $[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] A = [\begin{matrix} 3 & 3 & 5 \\ 1 & 0 & 1 \end{matrix}]$

(ii) $A [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 \\ 2 & 4 & 6 \end{matrix}]$

(iii) $[\begin{matrix} 4 \\ 1 \\ 3 \end{matrix}] A = [\begin{matrix} - 4 & 8 & 4 \\ - 1 & 2 & 1 \\ - 3 & 6 & 3 \end{matrix}]$

(iv) $[\begin{matrix} 2 & 1 & 3 \end{matrix}] [\begin{matrix} - 1 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 1 \end{matrix}] [\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}] = A$

(v) $[\begin{matrix} 2 & - 1 \\ 1 & 0 \\ - 3 & 4 \end{matrix}]$ A $= [\begin{matrix} - 1 & - 8 & - 10 \\ 1 & - 2 & - 5 \\ 9 & 22 & 15 \end{matrix}]$

(vi) A $= [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 \\ 2 & 4 & 6 \\ 11 & 10 & 9 \end{matrix}]$

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Solution

$(i) Let A = [\begin{matrix} x & y & z \\ a & b & c \end{matrix}] \Rightarrow [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} x & y & z \\ a & b & c \end{matrix}] = [\begin{matrix} 3 & 3 & 5 \\ 1 & 0 & 1 \end{matrix}] \Rightarrow [\begin{matrix} x + a & y + b & z + c \\ 0 + a & 0 + b & 0 + c \end{matrix}] = [\begin{matrix} 3 & 3 & 5 \\ 1 & 0 & 1 \end{matrix}] \Rightarrow [\begin{matrix} x + a & y + b & z + c \\ a & b & c \end{matrix}] = [\begin{matrix} 3 & 3 & 5 \\ 1 & 0 & 1 \end{matrix}] The corresponding elements of two equal matrices are equal . \Rightarrow x + a = 3 . . . (1) y + b = 3 . . . (2) z + c = 5 . . . (3) \Rightarrow a = 1, b = 0 and c = 1 Putting the value of a in eq . (1), we get x + 1 = 3 \Rightarrow x = 3 - 1 ∴ x = 2 Putting the value of b in eq . (2), we get y + b = 3 \Rightarrow y + 0 = 3 ∴ y = 3 Putting the value of c in eq . (3), we get z + 1 = 5 \Rightarrow z = 5 - 1 ∴ z = 4 ∴ A = [\begin{matrix} 2 & 3 & 4 \\ 1 & 0 & 1 \end{matrix}]$

$(ii) Let A = [\begin{matrix} w & x \\ y & z \end{matrix}] \Rightarrow A [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 \\ 2 & 4 & 6 \end{matrix}] \Rightarrow [\begin{matrix} w & x \\ y & z \end{matrix}] [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 \\ 2 & 4 & 6 \end{matrix}] \Rightarrow [\begin{matrix} w + 4 x & 2 w + 5 x & 3 w + 6 x \\ y + 4 z & 2 y + 5 z & 3 y + 6 z \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 \\ 2 & 4 & 6 \end{matrix}] The correspnding elements of two equal matrices are equal . ∴ 3 w + 6 x = - 9 . . . (1) y + 4 z = 2 y = 2 - 4 z . . . (2) w + 4 x = - 7 \Rightarrow w = - 7 - 4 x . . . (3) 2 y + 5 z = 4 . . . (4) Putting the value of w in eq . (1), we get 3 (- 7 - 4 x) + 6 x = - 9 \Rightarrow - 21 - 12 x + 6 x = - 9 \Rightarrow - 6 x = 12 \Rightarrow x = - 2 Putting the value of x in eq . (3), we get w = - 7 - 4 (- 2) \Rightarrow w = - 7 + 8 \Rightarrow w = 1 Putting the value of y in eq . (4), we get 2 (2 - 4 z) + 5 z = 4 \Rightarrow 4 - 8 z + 5 z = 4 \Rightarrow 4 - 3 z = 4 \Rightarrow - 3 z = 0 \Rightarrow z = 0 Putting the value of z in eq . (2), we get y = 2 - 4 (0) \Rightarrow y = 2 ∴ A = [\begin{matrix} 1 & - 2 \\ 2 & 0 \end{matrix}]$

$(iii) Let A = [\begin{matrix} x & y & z \end{matrix}] \Rightarrow [\begin{matrix} 4 \\ 1 \\ 3 \end{matrix}] [\begin{matrix} x & y & z \end{matrix}] = [\begin{matrix} - 4 & 8 & 4 \\ - 1 & 2 & 1 \\ - 3 & 6 & 3 \end{matrix}] \Rightarrow [\begin{matrix} 4 x & 4 y & 4 z \\ x & y & z \\ 3 x & 3 y & 3 z \end{matrix}] = [\begin{matrix} - 4 & 8 & 4 \\ - 1 & 2 & 1 \\ - 3 & 6 & 3 \end{matrix}] The corresponding elements of two equal matrices are equal . \Rightarrow 4 x = - 4 . . . (1) 4 y = 8 . . . (2) 4 z = 4 . . . (3) \Rightarrow x = - 1, y = 2 and z = 1 ∴ A = [\begin{matrix} - 1 & 2 & 1 \end{matrix}]$

$(iv) Let A = [\begin{matrix} x \end{matrix}] \Rightarrow [\begin{matrix} 2 & 1 & 3 \end{matrix}] [\begin{matrix} - 1 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 1 \end{matrix}] [\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}] = A \Rightarrow [\begin{matrix} 2 & 1 & 3 \end{matrix}] [\begin{matrix} - 1 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 1 \end{matrix}] [\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}] = [\begin{matrix} x \end{matrix}] \Rightarrow [\begin{matrix} - 2 - 1 + 0 & 0 + 1 + 3 & - 2 + 0 + 3 \end{matrix}] [\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}] = [\begin{matrix} x \end{matrix}] \Rightarrow [\begin{matrix} - 3 & 4 & 1 \end{matrix}] [\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}] = [\begin{matrix} x \end{matrix}] \Rightarrow [\begin{matrix} - 3 + 0 - 1 \end{matrix}] = [\begin{matrix} x \end{matrix}] \Rightarrow [\begin{matrix} - 4 \end{matrix}] = [\begin{matrix} x \end{matrix}] The corresponding elements of two equal matrices are equal . ∴ x = - 4 ∴ A = [- 4]$

$(v) [\begin{matrix} 2 & - 1 \\ 1 & 0 \\ - 3 & 4 \end{matrix}] A = [\begin{matrix} - 1 & - 8 & - 10 \\ 1 & - 2 & - 5 \\ 9 & 22 & 15 \end{matrix}] Let A = [\begin{matrix} x & y & z \\ a & b & c \end{matrix}] \Rightarrow [\begin{matrix} 2 & - 1 \\ 1 & 0 \\ - 3 & 4 \end{matrix}] [\begin{matrix} x & y & z \\ a & b & c \end{matrix}] = [\begin{matrix} - 1 & - 8 & - 10 \\ 1 & - 2 & - 5 \\ 9 & 22 & 15 \end{matrix}] \Rightarrow [\begin{matrix} 2 x - a & 2 y - b & 2 z - c \\ x & y & z \\ - 3 x + 4 a & - 3 y + 4 b & - 3 z + 4 c \end{matrix}] = [\begin{matrix} - 1 & - 8 & - 10 \\ 1 & - 2 & - 5 \\ 9 & 22 & 15 \end{matrix}] By comparing the elements of second row, we get x = 1, y = - 2, z = - 5 By comparing the elements of first row, we get 2 x - a = - 1 \Rightarrow 2 - a = - 1 \Rightarrow a = 3 Also, 2 y - b = - 8 \Rightarrow - 4 - b = - 8 \Rightarrow b = 4 And, 2 z - c = - 10 \Rightarrow - 10 - c = - 10 \Rightarrow c = 0 ∴ A = [\begin{matrix} 1 & - 2 & - 5 \\ 3 & 4 & 0 \end{matrix}]$

$(vi) A [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 \\ 2 & 4 & 6 \\ 11 & 10 & 9 \end{matrix}] Let A = [\begin{matrix} x & a \\ y & b \\ z & c \end{matrix}] \Rightarrow [\begin{matrix} x & a \\ y & b \\ z & c \end{matrix}] [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 \\ 2 & 4 & 6 \\ 11 & 10 & 9 \end{matrix}] \Rightarrow [\begin{matrix} x + 4 a & 2 x + 5 a & 3 x + 6 a \\ y + 4 b & 2 y + 5 b & 3 y + 6 b \\ z + 4 c & 2 z + 5 c & 3 z + 6 c \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 \\ 2 & 4 & 6 \\ 11 & 10 & 9 \end{matrix}] By comparing the corresponding elements, we get x + 4 a = - 7 and 2 x + 5 a = - 8 \Rightarrow a = - 2 and x = 1 Also, y + 4 b = 2 and 2 y + 5 b = 4 \Rightarrow b = 0 and y = 2 And, z + 4 c = 11 and 2 z + 5 c = 10 \Rightarrow c = 4 and z = - 5 ∴ A = [\begin{matrix} 1 & - 2 \\ 2 & 0 \\ - 5 & 4 \end{matrix}]$

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Similar questions

Q. Find the matrix A such that
(i)

[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] A = [\begin{matrix} 3 & 3 & 5 \\ 1 & 0 & 1 \end{matrix}]

A [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 \\ 2 & 4 & 6 \end{matrix}]

[\begin{matrix} 4 \\ 1 \\ 3 \end{matrix}] A = [\begin{matrix} - 4 & 8 & 4 \\ - 1 & 2 & 1 \\ - 3 & 6 & 3 \end{matrix}]

[\begin{matrix} 2 & 1 & 3 \end{matrix}] [\begin{matrix} - 1 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 1 \end{matrix}] [\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}] = A

Q. Find the matrix A such that
(i)

[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] A = [\begin{matrix} 3 & 3 & 5 \\ 1 & 0 & 1 \end{matrix}]

A [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 \\ 2 & 4 & 6 \end{matrix}]

Q. Find the matrix

X

X [\begin{matrix} 1 & 2 & 3 4 & 5 & 6 \end{matrix}] = [\begin{matrix} - 7 & - 8 & - 9 2 & 4 & 6 \end{matrix}]

[\begin{matrix} 2 & 1 & 3 \end{matrix}] (\begin{matrix} - 1 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 1 \end{matrix}) (\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}) = A

, then write the order of matrix A.

(10)^{9} + 2 (11)^{1} (10)^{8} + 3 (11)^{2} (10)^{7} + \dots \dots + 10 (11)^{9} = k (10)^{9}

, then k is equal to

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