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Page 1 Solved Examples on Matrices JEE Mains Q1. If ?? = [ ?? ?? ?? ?? ] , ?? ? ?? , then ?? ?? ?? equals (a) [ ?? ?? ?? ?? ] (b) [ ?? ?? ?? ?? ] (c) [ ?? ?? ?? ?? ] (d) [ ?? ?? ?? ?? ] Ans: (a) ?? 2 = [ ?? 0 0 ?? ] [ ?? 0 0 ?? ] = [ -1 0 0 -1 ] , ?? 4 = ?? 2 · ?? 2 = [ -1 0 0 -1 ] [ -1 0 0 -1 ] = [ 1 0 0 1 ] = ?? ; ( ?? 4 ) ?? = ?? ?? = ?? = [ 1 0 0 1 ] Q2. For the matrix ?? = [ ?? ?? ?? ?? ?? ?? ?? ?? ?? ], which of the following is correct (a) ?? ?? + ?? ?? ?? - ?? = ?? (b) ?? ?? - ?? ?? ?? - ?? = ?? (c) ?? ?? + ?? ?? ?? - ?? = ?? (d) ?? ?? - ?? ?? + ?? = ?? Ans: (b) ?? 2 = ?? · ?? = [ 1 1 0 1 2 1 2 1 0 ] [ 1 1 0 1 2 1 2 1 0 ] = [ 2 3 1 5 6 2 3 4 1 ] , ?? 3 = ?? · 2 ?? = [ 2 3 1 5 6 2 3 4 1 ] [ 1 1 0 1 2 1 2 1 0 ] = [ 7 9 3 15 19 6 9 12 4 ] ?? 3 - 3 · ?? 2 = [ 7 9 3 15 19 6 9 12 4 ] - [ 6 9 3 15 18 6 9 12 3 ] = [ 1 0 0 0 1 0 0 0 1 ] = ?? ? ?? 3 - 3?? 2 - ?? = 0 Q3. The matrix [ ?? ?? -?? -?? ?? ?? ?? -?? -?? ] is non singular if (a) ?? ? -?? (b) ?? ? ?? (c) ?? ? ?? (d) ?? ? -?? Ans: (a) The given matrix ?? = [ 2 ?? -4 -1 3 4 1 -2 -3 ] is non singular If |?? | ? 0 Page 2 Solved Examples on Matrices JEE Mains Q1. If ?? = [ ?? ?? ?? ?? ] , ?? ? ?? , then ?? ?? ?? equals (a) [ ?? ?? ?? ?? ] (b) [ ?? ?? ?? ?? ] (c) [ ?? ?? ?? ?? ] (d) [ ?? ?? ?? ?? ] Ans: (a) ?? 2 = [ ?? 0 0 ?? ] [ ?? 0 0 ?? ] = [ -1 0 0 -1 ] , ?? 4 = ?? 2 · ?? 2 = [ -1 0 0 -1 ] [ -1 0 0 -1 ] = [ 1 0 0 1 ] = ?? ; ( ?? 4 ) ?? = ?? ?? = ?? = [ 1 0 0 1 ] Q2. For the matrix ?? = [ ?? ?? ?? ?? ?? ?? ?? ?? ?? ], which of the following is correct (a) ?? ?? + ?? ?? ?? - ?? = ?? (b) ?? ?? - ?? ?? ?? - ?? = ?? (c) ?? ?? + ?? ?? ?? - ?? = ?? (d) ?? ?? - ?? ?? + ?? = ?? Ans: (b) ?? 2 = ?? · ?? = [ 1 1 0 1 2 1 2 1 0 ] [ 1 1 0 1 2 1 2 1 0 ] = [ 2 3 1 5 6 2 3 4 1 ] , ?? 3 = ?? · 2 ?? = [ 2 3 1 5 6 2 3 4 1 ] [ 1 1 0 1 2 1 2 1 0 ] = [ 7 9 3 15 19 6 9 12 4 ] ?? 3 - 3 · ?? 2 = [ 7 9 3 15 19 6 9 12 4 ] - [ 6 9 3 15 18 6 9 12 3 ] = [ 1 0 0 0 1 0 0 0 1 ] = ?? ? ?? 3 - 3?? 2 - ?? = 0 Q3. The matrix [ ?? ?? -?? -?? ?? ?? ?? -?? -?? ] is non singular if (a) ?? ? -?? (b) ?? ? ?? (c) ?? ? ?? (d) ?? ? -?? Ans: (a) The given matrix ?? = [ 2 ?? -4 -1 3 4 1 -2 -3 ] is non singular If |?? | ? 0 ? |?? | = | 2 ?? -4 -1 3 4 1 -2 -3 | ? 0 ? | 1 ?? + 3 0 -1 3 4 1 -2 -3 | ? 0[?? 1 ? ?? 1 + ?? 2 ] ? |?? | = [ 1 ?? + 3 0 0 1 1 0 -?? - 5 -3 ] ? 0 , [ ?? 2 ? ?? 2 + ?? 3 ?? 3 ? ?? 3 - ?? 1 ] ? 1( -3 + ?? + 5)? 0 ? ?? + 2 ? 0 ? ?? ? -2 Q4. Let ?? = [ ?? -?? ?? ?? ?? -?? ?? ?? ?? ] and ???? . ?? = [ ?? ?? ?? -?? ?? ?? ?? -?? ?? ]. If ?? is the inverse of matrix ?? , then ?? is (a) 5 (b) -1 (c) 2 (d) -2 Ans: (a) We have, ?? = [ 1 -1 1 -2 1 -3 1 1 1 ] , ? |?? | = 1( 4)+ 1( 5)+ 1( 1)= 10 and adj ( ?? )= [ 4 2 2 -5 0 5 1 -2 3 ] Then ?? -1 = 1 10 [ 4 2 2 -5 0 5 1 -2 3 ] According to question, ?? is the inverse of matrix ?? . Hence ?? = 5 Q5. If ?? = [ ?? ?? ?? ?? ?? ?? ?? ?? ?? ] and ?? -?? = [ ?? /?? -?? /?? ?? /?? -?? ?? ?? ?? /?? -?? /?? ?? /?? ], then (a) ?? = ?? , ?? = -?? (b) ?? = -?? , ?? = ?? (c) ?? = ?? , ?? = -?? /?? (d) ?? = ?? /?? , ?? = ?? ?? Ans: (a) We know AA -1 = I, hence by solving it we can obtain the values of x and y. We have [ 1 0 0 0 1 0 0 0 1 ] = AA -1 = [ 0 1 2 1 2 3 3 ?? 1 ] [ 1/2 -1/2 1/2 -4 3 ?? 5/2 -3/2 1/2 ] = [ 1 0 ?? + 1 0 1 2( ?? + 1) 4( 1 - ?? ) 3( ?? - 1) 2 + ???? ] ? 1 - ?? = 0, ?? - 1 = 0; ?? + 1 = 0, ?? + 1 = 0,2 + ???? = 1; ? ?? = 1, ?? = -1 Q6. If [ ?? ?? ?? ?? ] ?? [ -?? ?? ?? -?? ] = [ ?? ?? ?? ?? ], then matrix ?? equals: (a) [ ?? ?? -???? -?? ] (b) [ ?? ?? ?? ?? ] (c) [ ?? ?? ???? ?? ] (d) [ ?? ?? ???? ?? ] Page 3 Solved Examples on Matrices JEE Mains Q1. If ?? = [ ?? ?? ?? ?? ] , ?? ? ?? , then ?? ?? ?? equals (a) [ ?? ?? ?? ?? ] (b) [ ?? ?? ?? ?? ] (c) [ ?? ?? ?? ?? ] (d) [ ?? ?? ?? ?? ] Ans: (a) ?? 2 = [ ?? 0 0 ?? ] [ ?? 0 0 ?? ] = [ -1 0 0 -1 ] , ?? 4 = ?? 2 · ?? 2 = [ -1 0 0 -1 ] [ -1 0 0 -1 ] = [ 1 0 0 1 ] = ?? ; ( ?? 4 ) ?? = ?? ?? = ?? = [ 1 0 0 1 ] Q2. For the matrix ?? = [ ?? ?? ?? ?? ?? ?? ?? ?? ?? ], which of the following is correct (a) ?? ?? + ?? ?? ?? - ?? = ?? (b) ?? ?? - ?? ?? ?? - ?? = ?? (c) ?? ?? + ?? ?? ?? - ?? = ?? (d) ?? ?? - ?? ?? + ?? = ?? Ans: (b) ?? 2 = ?? · ?? = [ 1 1 0 1 2 1 2 1 0 ] [ 1 1 0 1 2 1 2 1 0 ] = [ 2 3 1 5 6 2 3 4 1 ] , ?? 3 = ?? · 2 ?? = [ 2 3 1 5 6 2 3 4 1 ] [ 1 1 0 1 2 1 2 1 0 ] = [ 7 9 3 15 19 6 9 12 4 ] ?? 3 - 3 · ?? 2 = [ 7 9 3 15 19 6 9 12 4 ] - [ 6 9 3 15 18 6 9 12 3 ] = [ 1 0 0 0 1 0 0 0 1 ] = ?? ? ?? 3 - 3?? 2 - ?? = 0 Q3. The matrix [ ?? ?? -?? -?? ?? ?? ?? -?? -?? ] is non singular if (a) ?? ? -?? (b) ?? ? ?? (c) ?? ? ?? (d) ?? ? -?? Ans: (a) The given matrix ?? = [ 2 ?? -4 -1 3 4 1 -2 -3 ] is non singular If |?? | ? 0 ? |?? | = | 2 ?? -4 -1 3 4 1 -2 -3 | ? 0 ? | 1 ?? + 3 0 -1 3 4 1 -2 -3 | ? 0[?? 1 ? ?? 1 + ?? 2 ] ? |?? | = [ 1 ?? + 3 0 0 1 1 0 -?? - 5 -3 ] ? 0 , [ ?? 2 ? ?? 2 + ?? 3 ?? 3 ? ?? 3 - ?? 1 ] ? 1( -3 + ?? + 5)? 0 ? ?? + 2 ? 0 ? ?? ? -2 Q4. Let ?? = [ ?? -?? ?? ?? ?? -?? ?? ?? ?? ] and ???? . ?? = [ ?? ?? ?? -?? ?? ?? ?? -?? ?? ]. If ?? is the inverse of matrix ?? , then ?? is (a) 5 (b) -1 (c) 2 (d) -2 Ans: (a) We have, ?? = [ 1 -1 1 -2 1 -3 1 1 1 ] , ? |?? | = 1( 4)+ 1( 5)+ 1( 1)= 10 and adj ( ?? )= [ 4 2 2 -5 0 5 1 -2 3 ] Then ?? -1 = 1 10 [ 4 2 2 -5 0 5 1 -2 3 ] According to question, ?? is the inverse of matrix ?? . Hence ?? = 5 Q5. If ?? = [ ?? ?? ?? ?? ?? ?? ?? ?? ?? ] and ?? -?? = [ ?? /?? -?? /?? ?? /?? -?? ?? ?? ?? /?? -?? /?? ?? /?? ], then (a) ?? = ?? , ?? = -?? (b) ?? = -?? , ?? = ?? (c) ?? = ?? , ?? = -?? /?? (d) ?? = ?? /?? , ?? = ?? ?? Ans: (a) We know AA -1 = I, hence by solving it we can obtain the values of x and y. We have [ 1 0 0 0 1 0 0 0 1 ] = AA -1 = [ 0 1 2 1 2 3 3 ?? 1 ] [ 1/2 -1/2 1/2 -4 3 ?? 5/2 -3/2 1/2 ] = [ 1 0 ?? + 1 0 1 2( ?? + 1) 4( 1 - ?? ) 3( ?? - 1) 2 + ???? ] ? 1 - ?? = 0, ?? - 1 = 0; ?? + 1 = 0, ?? + 1 = 0,2 + ???? = 1; ? ?? = 1, ?? = -1 Q6. If [ ?? ?? ?? ?? ] ?? [ -?? ?? ?? -?? ] = [ ?? ?? ?? ?? ], then matrix ?? equals: (a) [ ?? ?? -???? -?? ] (b) [ ?? ?? ?? ?? ] (c) [ ?? ?? ???? ?? ] (d) [ ?? ?? ???? ?? ] Ans: (a) We know that if ?????? = ?? , then ?? = ?? -1 ?? -1 = ( ???? ) -1 . In this case ???? = [ -3 2 5 -3 ] [ 2 1 7 4 ] = [ 8 5 -11 -7 ] ; ? ?? = [ 8 5 -11 -7 ] -1 = [ 7 5 -11 -8 ] Q7. If ?? is a skew symmetric matrix such that ?? ? ?? = ?? , then ?? ?? ?? -?? ( ?? ? ?? ) is equal to (A) -?? ? (B) I (C) - I (D) ?? ? Ans: (D) ?? is skew symmetric matrix ? ?? ? ?? = ?? ? ?? ? = -?? ? ( ???? ) 2 = ( ?? ) 2 ? ?? ? ?? = -?? 2 = ?? Taking square of both sides ?? 4 = ?? ? ?? 4?? = ?? ( ? -?? 2 = ?? are ?? = -?? ? )\ ?? 3 ?? 4?? = ?? 3 ?? ?? 4?? +3 = -?? 2 ( -?? ) ?? ?? 4( ?? +1) -1 = ( -?? ) ?? = ?? ? ?? = ?? ?? 4?? - 1 ? ?? ? ?? 4?? -1 = ?? ? Q8. ?? and ?? are ?? × ?? matrices satisfying det ?? = ?????? ?? and ???? ( ?? )= ???? ( ?? ) , further ?? ?? - ?? ?? + ???? ?? = ?? and ?? ?? - ???? + ???? = ?? , then ?? is equal to (A) 3 (B) 11 (C) -11 (D) 14 Ans: (D) |?? | = |?? |?? 2×2 '?? 2×2 Tr ( ?? )= Tr ( ?? ) ?? 2 - 3?? + 14?? = 0 and ?? 2 - ???? + ???? = 0 if ?? = ?? , |?? | = |?? | Tr |?? | = Tr ( ?? ) satisfied so ?? 2 - ???? + ???? = 0 ?? = 14( ? A's order ' = 2 × 2) Q9. If ?? = [ ?? ?? -?? ?? ] is an orthogonal matrix, where ?? , ?? and the roots other than the common root of the equations ?? ?? - ???? + ?? = ?? &?? ?? + ???? - ?? = ?? , then (A) ?? = ± ?? v?? , ?? = ± ?? v?? (B) ?? = ?? , ?? = ± ?? v?? Page 4 Solved Examples on Matrices JEE Mains Q1. If ?? = [ ?? ?? ?? ?? ] , ?? ? ?? , then ?? ?? ?? equals (a) [ ?? ?? ?? ?? ] (b) [ ?? ?? ?? ?? ] (c) [ ?? ?? ?? ?? ] (d) [ ?? ?? ?? ?? ] Ans: (a) ?? 2 = [ ?? 0 0 ?? ] [ ?? 0 0 ?? ] = [ -1 0 0 -1 ] , ?? 4 = ?? 2 · ?? 2 = [ -1 0 0 -1 ] [ -1 0 0 -1 ] = [ 1 0 0 1 ] = ?? ; ( ?? 4 ) ?? = ?? ?? = ?? = [ 1 0 0 1 ] Q2. For the matrix ?? = [ ?? ?? ?? ?? ?? ?? ?? ?? ?? ], which of the following is correct (a) ?? ?? + ?? ?? ?? - ?? = ?? (b) ?? ?? - ?? ?? ?? - ?? = ?? (c) ?? ?? + ?? ?? ?? - ?? = ?? (d) ?? ?? - ?? ?? + ?? = ?? Ans: (b) ?? 2 = ?? · ?? = [ 1 1 0 1 2 1 2 1 0 ] [ 1 1 0 1 2 1 2 1 0 ] = [ 2 3 1 5 6 2 3 4 1 ] , ?? 3 = ?? · 2 ?? = [ 2 3 1 5 6 2 3 4 1 ] [ 1 1 0 1 2 1 2 1 0 ] = [ 7 9 3 15 19 6 9 12 4 ] ?? 3 - 3 · ?? 2 = [ 7 9 3 15 19 6 9 12 4 ] - [ 6 9 3 15 18 6 9 12 3 ] = [ 1 0 0 0 1 0 0 0 1 ] = ?? ? ?? 3 - 3?? 2 - ?? = 0 Q3. The matrix [ ?? ?? -?? -?? ?? ?? ?? -?? -?? ] is non singular if (a) ?? ? -?? (b) ?? ? ?? (c) ?? ? ?? (d) ?? ? -?? Ans: (a) The given matrix ?? = [ 2 ?? -4 -1 3 4 1 -2 -3 ] is non singular If |?? | ? 0 ? |?? | = | 2 ?? -4 -1 3 4 1 -2 -3 | ? 0 ? | 1 ?? + 3 0 -1 3 4 1 -2 -3 | ? 0[?? 1 ? ?? 1 + ?? 2 ] ? |?? | = [ 1 ?? + 3 0 0 1 1 0 -?? - 5 -3 ] ? 0 , [ ?? 2 ? ?? 2 + ?? 3 ?? 3 ? ?? 3 - ?? 1 ] ? 1( -3 + ?? + 5)? 0 ? ?? + 2 ? 0 ? ?? ? -2 Q4. Let ?? = [ ?? -?? ?? ?? ?? -?? ?? ?? ?? ] and ???? . ?? = [ ?? ?? ?? -?? ?? ?? ?? -?? ?? ]. If ?? is the inverse of matrix ?? , then ?? is (a) 5 (b) -1 (c) 2 (d) -2 Ans: (a) We have, ?? = [ 1 -1 1 -2 1 -3 1 1 1 ] , ? |?? | = 1( 4)+ 1( 5)+ 1( 1)= 10 and adj ( ?? )= [ 4 2 2 -5 0 5 1 -2 3 ] Then ?? -1 = 1 10 [ 4 2 2 -5 0 5 1 -2 3 ] According to question, ?? is the inverse of matrix ?? . Hence ?? = 5 Q5. If ?? = [ ?? ?? ?? ?? ?? ?? ?? ?? ?? ] and ?? -?? = [ ?? /?? -?? /?? ?? /?? -?? ?? ?? ?? /?? -?? /?? ?? /?? ], then (a) ?? = ?? , ?? = -?? (b) ?? = -?? , ?? = ?? (c) ?? = ?? , ?? = -?? /?? (d) ?? = ?? /?? , ?? = ?? ?? Ans: (a) We know AA -1 = I, hence by solving it we can obtain the values of x and y. We have [ 1 0 0 0 1 0 0 0 1 ] = AA -1 = [ 0 1 2 1 2 3 3 ?? 1 ] [ 1/2 -1/2 1/2 -4 3 ?? 5/2 -3/2 1/2 ] = [ 1 0 ?? + 1 0 1 2( ?? + 1) 4( 1 - ?? ) 3( ?? - 1) 2 + ???? ] ? 1 - ?? = 0, ?? - 1 = 0; ?? + 1 = 0, ?? + 1 = 0,2 + ???? = 1; ? ?? = 1, ?? = -1 Q6. If [ ?? ?? ?? ?? ] ?? [ -?? ?? ?? -?? ] = [ ?? ?? ?? ?? ], then matrix ?? equals: (a) [ ?? ?? -???? -?? ] (b) [ ?? ?? ?? ?? ] (c) [ ?? ?? ???? ?? ] (d) [ ?? ?? ???? ?? ] Ans: (a) We know that if ?????? = ?? , then ?? = ?? -1 ?? -1 = ( ???? ) -1 . In this case ???? = [ -3 2 5 -3 ] [ 2 1 7 4 ] = [ 8 5 -11 -7 ] ; ? ?? = [ 8 5 -11 -7 ] -1 = [ 7 5 -11 -8 ] Q7. If ?? is a skew symmetric matrix such that ?? ? ?? = ?? , then ?? ?? ?? -?? ( ?? ? ?? ) is equal to (A) -?? ? (B) I (C) - I (D) ?? ? Ans: (D) ?? is skew symmetric matrix ? ?? ? ?? = ?? ? ?? ? = -?? ? ( ???? ) 2 = ( ?? ) 2 ? ?? ? ?? = -?? 2 = ?? Taking square of both sides ?? 4 = ?? ? ?? 4?? = ?? ( ? -?? 2 = ?? are ?? = -?? ? )\ ?? 3 ?? 4?? = ?? 3 ?? ?? 4?? +3 = -?? 2 ( -?? ) ?? ?? 4( ?? +1) -1 = ( -?? ) ?? = ?? ? ?? = ?? ?? 4?? - 1 ? ?? ? ?? 4?? -1 = ?? ? Q8. ?? and ?? are ?? × ?? matrices satisfying det ?? = ?????? ?? and ???? ( ?? )= ???? ( ?? ) , further ?? ?? - ?? ?? + ???? ?? = ?? and ?? ?? - ???? + ???? = ?? , then ?? is equal to (A) 3 (B) 11 (C) -11 (D) 14 Ans: (D) |?? | = |?? |?? 2×2 '?? 2×2 Tr ( ?? )= Tr ( ?? ) ?? 2 - 3?? + 14?? = 0 and ?? 2 - ???? + ???? = 0 if ?? = ?? , |?? | = |?? | Tr |?? | = Tr ( ?? ) satisfied so ?? 2 - ???? + ???? = 0 ?? = 14( ? A's order ' = 2 × 2) Q9. If ?? = [ ?? ?? -?? ?? ] is an orthogonal matrix, where ?? , ?? and the roots other than the common root of the equations ?? ?? - ???? + ?? = ?? &?? ?? + ???? - ?? = ?? , then (A) ?? = ± ?? v?? , ?? = ± ?? v?? (B) ?? = ?? , ?? = ± ?? v?? (C) ?? = ± ?? v?? , ?? = ?? (D) None of these Ans: (C) ?? = [ ?? ?? -?? ?? ] Since, ?? is orthogonal matrix So, AA ' = A ' A = I n AA -1 = [ ?? ?? -?? ?? ] [ ?? -?? ?? ?? ] = [ ?? 2 + ?? 2 -???? + ???? -???? + ???? ?? 2 + ?? 2 ] [ ?? 2 + ?? 2 0 0 ?? 2 + ?? 2 ] = [ 1 0 0 1 ] ?? 2 + ?? 2 = 1 ?? 2 - ???? + ?? = 0&?? 2 + ???? - ?? = 0 Sum of both equation for common roots ?? 2 - ???? + ?? + ?? 2 + ???? - ?? = 0 2?? 2 = 0 ? ?? = 0 So if ?? is roots of ?? 2 - ???? + ?? = 0 and ?? is roots of ?? 2 + px - q = 0 ? ?? + 0 = ?? and ?? ( 0)= ?? = 0 ?? + 0 = -?? and ?? ( 0)= -?? = 0 ? ?? = ?? and ?? = -?? In (i) equation a 2 + b 2 = 1 ? p 2 + ( -p) 2 - 1 ? 2p 2 = 1 ? p = ± 1 v2 , q = 0 Q10. ?? is a square matrix of order ?? and ( ?????? ?? )= ?? . If ?????? ( ???? )= ???? ; where ?? ? ?? , then possible value of ?? is (A) 3 (B) 5 (C) 2 (D) 7 Ans: (A) ( det ?? )= 3 ( det ???? )= 81 if A's order = n × n then ( det ?? · ?? )= ?? ?? ( det ?? )= ?? × 3 = 81 ?? ?? = 81 3 = 27 ?? ?? = 27 = 3 ?? ? ?? ? ?? So, n = 3 Page 5 Solved Examples on Matrices JEE Mains Q1. If ?? = [ ?? ?? ?? ?? ] , ?? ? ?? , then ?? ?? ?? equals (a) [ ?? ?? ?? ?? ] (b) [ ?? ?? ?? ?? ] (c) [ ?? ?? ?? ?? ] (d) [ ?? ?? ?? ?? ] Ans: (a) ?? 2 = [ ?? 0 0 ?? ] [ ?? 0 0 ?? ] = [ -1 0 0 -1 ] , ?? 4 = ?? 2 · ?? 2 = [ -1 0 0 -1 ] [ -1 0 0 -1 ] = [ 1 0 0 1 ] = ?? ; ( ?? 4 ) ?? = ?? ?? = ?? = [ 1 0 0 1 ] Q2. For the matrix ?? = [ ?? ?? ?? ?? ?? ?? ?? ?? ?? ], which of the following is correct (a) ?? ?? + ?? ?? ?? - ?? = ?? (b) ?? ?? - ?? ?? ?? - ?? = ?? (c) ?? ?? + ?? ?? ?? - ?? = ?? (d) ?? ?? - ?? ?? + ?? = ?? Ans: (b) ?? 2 = ?? · ?? = [ 1 1 0 1 2 1 2 1 0 ] [ 1 1 0 1 2 1 2 1 0 ] = [ 2 3 1 5 6 2 3 4 1 ] , ?? 3 = ?? · 2 ?? = [ 2 3 1 5 6 2 3 4 1 ] [ 1 1 0 1 2 1 2 1 0 ] = [ 7 9 3 15 19 6 9 12 4 ] ?? 3 - 3 · ?? 2 = [ 7 9 3 15 19 6 9 12 4 ] - [ 6 9 3 15 18 6 9 12 3 ] = [ 1 0 0 0 1 0 0 0 1 ] = ?? ? ?? 3 - 3?? 2 - ?? = 0 Q3. The matrix [ ?? ?? -?? -?? ?? ?? ?? -?? -?? ] is non singular if (a) ?? ? -?? (b) ?? ? ?? (c) ?? ? ?? (d) ?? ? -?? Ans: (a) The given matrix ?? = [ 2 ?? -4 -1 3 4 1 -2 -3 ] is non singular If |?? | ? 0 ? |?? | = | 2 ?? -4 -1 3 4 1 -2 -3 | ? 0 ? | 1 ?? + 3 0 -1 3 4 1 -2 -3 | ? 0[?? 1 ? ?? 1 + ?? 2 ] ? |?? | = [ 1 ?? + 3 0 0 1 1 0 -?? - 5 -3 ] ? 0 , [ ?? 2 ? ?? 2 + ?? 3 ?? 3 ? ?? 3 - ?? 1 ] ? 1( -3 + ?? + 5)? 0 ? ?? + 2 ? 0 ? ?? ? -2 Q4. Let ?? = [ ?? -?? ?? ?? ?? -?? ?? ?? ?? ] and ???? . ?? = [ ?? ?? ?? -?? ?? ?? ?? -?? ?? ]. If ?? is the inverse of matrix ?? , then ?? is (a) 5 (b) -1 (c) 2 (d) -2 Ans: (a) We have, ?? = [ 1 -1 1 -2 1 -3 1 1 1 ] , ? |?? | = 1( 4)+ 1( 5)+ 1( 1)= 10 and adj ( ?? )= [ 4 2 2 -5 0 5 1 -2 3 ] Then ?? -1 = 1 10 [ 4 2 2 -5 0 5 1 -2 3 ] According to question, ?? is the inverse of matrix ?? . Hence ?? = 5 Q5. If ?? = [ ?? ?? ?? ?? ?? ?? ?? ?? ?? ] and ?? -?? = [ ?? /?? -?? /?? ?? /?? -?? ?? ?? ?? /?? -?? /?? ?? /?? ], then (a) ?? = ?? , ?? = -?? (b) ?? = -?? , ?? = ?? (c) ?? = ?? , ?? = -?? /?? (d) ?? = ?? /?? , ?? = ?? ?? Ans: (a) We know AA -1 = I, hence by solving it we can obtain the values of x and y. We have [ 1 0 0 0 1 0 0 0 1 ] = AA -1 = [ 0 1 2 1 2 3 3 ?? 1 ] [ 1/2 -1/2 1/2 -4 3 ?? 5/2 -3/2 1/2 ] = [ 1 0 ?? + 1 0 1 2( ?? + 1) 4( 1 - ?? ) 3( ?? - 1) 2 + ???? ] ? 1 - ?? = 0, ?? - 1 = 0; ?? + 1 = 0, ?? + 1 = 0,2 + ???? = 1; ? ?? = 1, ?? = -1 Q6. If [ ?? ?? ?? ?? ] ?? [ -?? ?? ?? -?? ] = [ ?? ?? ?? ?? ], then matrix ?? equals: (a) [ ?? ?? -???? -?? ] (b) [ ?? ?? ?? ?? ] (c) [ ?? ?? ???? ?? ] (d) [ ?? ?? ???? ?? ] Ans: (a) We know that if ?????? = ?? , then ?? = ?? -1 ?? -1 = ( ???? ) -1 . In this case ???? = [ -3 2 5 -3 ] [ 2 1 7 4 ] = [ 8 5 -11 -7 ] ; ? ?? = [ 8 5 -11 -7 ] -1 = [ 7 5 -11 -8 ] Q7. If ?? is a skew symmetric matrix such that ?? ? ?? = ?? , then ?? ?? ?? -?? ( ?? ? ?? ) is equal to (A) -?? ? (B) I (C) - I (D) ?? ? Ans: (D) ?? is skew symmetric matrix ? ?? ? ?? = ?? ? ?? ? = -?? ? ( ???? ) 2 = ( ?? ) 2 ? ?? ? ?? = -?? 2 = ?? Taking square of both sides ?? 4 = ?? ? ?? 4?? = ?? ( ? -?? 2 = ?? are ?? = -?? ? )\ ?? 3 ?? 4?? = ?? 3 ?? ?? 4?? +3 = -?? 2 ( -?? ) ?? ?? 4( ?? +1) -1 = ( -?? ) ?? = ?? ? ?? = ?? ?? 4?? - 1 ? ?? ? ?? 4?? -1 = ?? ? Q8. ?? and ?? are ?? × ?? matrices satisfying det ?? = ?????? ?? and ???? ( ?? )= ???? ( ?? ) , further ?? ?? - ?? ?? + ???? ?? = ?? and ?? ?? - ???? + ???? = ?? , then ?? is equal to (A) 3 (B) 11 (C) -11 (D) 14 Ans: (D) |?? | = |?? |?? 2×2 '?? 2×2 Tr ( ?? )= Tr ( ?? ) ?? 2 - 3?? + 14?? = 0 and ?? 2 - ???? + ???? = 0 if ?? = ?? , |?? | = |?? | Tr |?? | = Tr ( ?? ) satisfied so ?? 2 - ???? + ???? = 0 ?? = 14( ? A's order ' = 2 × 2) Q9. If ?? = [ ?? ?? -?? ?? ] is an orthogonal matrix, where ?? , ?? and the roots other than the common root of the equations ?? ?? - ???? + ?? = ?? &?? ?? + ???? - ?? = ?? , then (A) ?? = ± ?? v?? , ?? = ± ?? v?? (B) ?? = ?? , ?? = ± ?? v?? (C) ?? = ± ?? v?? , ?? = ?? (D) None of these Ans: (C) ?? = [ ?? ?? -?? ?? ] Since, ?? is orthogonal matrix So, AA ' = A ' A = I n AA -1 = [ ?? ?? -?? ?? ] [ ?? -?? ?? ?? ] = [ ?? 2 + ?? 2 -???? + ???? -???? + ???? ?? 2 + ?? 2 ] [ ?? 2 + ?? 2 0 0 ?? 2 + ?? 2 ] = [ 1 0 0 1 ] ?? 2 + ?? 2 = 1 ?? 2 - ???? + ?? = 0&?? 2 + ???? - ?? = 0 Sum of both equation for common roots ?? 2 - ???? + ?? + ?? 2 + ???? - ?? = 0 2?? 2 = 0 ? ?? = 0 So if ?? is roots of ?? 2 - ???? + ?? = 0 and ?? is roots of ?? 2 + px - q = 0 ? ?? + 0 = ?? and ?? ( 0)= ?? = 0 ?? + 0 = -?? and ?? ( 0)= -?? = 0 ? ?? = ?? and ?? = -?? In (i) equation a 2 + b 2 = 1 ? p 2 + ( -p) 2 - 1 ? 2p 2 = 1 ? p = ± 1 v2 , q = 0 Q10. ?? is a square matrix of order ?? and ( ?????? ?? )= ?? . If ?????? ( ???? )= ???? ; where ?? ? ?? , then possible value of ?? is (A) 3 (B) 5 (C) 2 (D) 7 Ans: (A) ( det ?? )= 3 ( det ???? )= 81 if A's order = n × n then ( det ?? · ?? )= ?? ?? ( det ?? )= ?? × 3 = 81 ?? ?? = 81 3 = 27 ?? ?? = 27 = 3 ?? ? ?? ? ?? So, n = 3 Q12. If ?? = [ ?? ?? ?? ?? ] and ?? ( ?? )= ?? +?? ?? -?? , then ?? ( ?? ) is (A) [ ?? ?? ?? ?? ] (B) [ -?? -?? -?? -?? ] (C) [ ?? ?? ?? ?? ] (D) None of these Ans: (B) ?? = [ 1 2 2 1 ] , ?? ( ?? )= 1 + ?? 1 - ?? ?? + ?? = [ 1 0 0 1 ] + [ 1 2 2 1 ] = [ 2 2 2 2 ] = 2 [ 1 1 1 1 ] ?? - ?? = [ 1 0 0 1 ] - [ 1 2 2 1 ] = [ 0 -2 -2 0 ] = -2 [ 0 1 1 0 ] ( ?? - ?? ) -1 = 1 det ( ?? - ?? ) [ 0 2 2 0 ] det ( ?? - ?? )= | 0 -2 -2 0 | = 0 - ( -2) ( -2)= -4 ?? ( ?? )= ( ?? + ?? ) ( ?? - ?? ) = 2 -4 [ 1 1 1 1 ] [ 0 2 2 0 ] = - 1 2 [ 0 + 2 2 + 0 0 + 2 2 + 0 ] = - 2 2 [ 1 1 1 1 ] = [ -1 -1 -1 -1 ] Q13. If ?? is a two-rowed matrix satisfying ?? ? = ?? -?? , then ?? is (A) [ ?????? ?? -?????? ?? -?????? ?? ?????? ?? ] (B) [ ?????? ?? ?????? ?? -?????? ?? ?????? ?? ] (C) [ -?????? ?? ?????? ?? ?????? ?? -?????? ?? ] (D) None of these Ans: (B) ?? ? = ?? -1 Assume ?? = [ ?? 1 ?? 2 ?? 3 ?? 4 ] , ?? ?? = [ ?? 1 ?? 3 ?? 2 ?? 4 ] ? |?? | = ?? 1 ?? 4 - ?? 2 ?? 3 ?? -1 = 1 |?? | (adj ?? )= 1 |?? | [ ?? 4 -?? 3 -?? 2 ?? 1 ] ?? = 1 |?? | [ ?? 4 -?? 2 -?? 3 ?? 1 ] |?? | = ?? 1 ?? 4 - ?? 2 ?? 3 [ ?? 1 ?? 3 ?? 2 ?? 4 ] = 1 |?? | [ ?? 4 -?? 2 -?? 3 ?? 1 ] |?? | = 1 ?? 2 = -?? 3 Only option (B) [ cos ?? sin ?? -sin ?? cos ?? ] is correct.Read More
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Additional Information about Matrices Solved Examples for JEE Preparation
Matrices Solved Examples Free PDF Download
The Matrices Solved Examples is an invaluable resource that delves deep into the core of the JEE exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
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and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Matrices Solved Examples now and kickstart your journey towards success in the JEE exam.
Importance of Matrices Solved Examples
The importance of Matrices Solved Examples cannot be overstated, especially for JEE aspirants.
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Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Matrices Solved Examples Notes
Matrices Solved Examples Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Matrices Solved Examples.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
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Whether you're a beginner or aiming for advanced proficiency, Matrices Solved Examples Notes on EduRev are your ultimate resource for success.
Matrices Solved Examples JEE Questions
The "Matrices Solved Examples JEE Questions" guide is a valuable resource for all aspiring students preparing for the
JEE exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Matrices Solved Examples on the App
Students of JEE can study Matrices Solved Examples alongwith tests & analysis from the EduRev app,
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The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Matrices Solved Examples is prepared as per the latest JEE syllabus.
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