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Page 1 Solved Examples on Matrices JEE Mains Q1. If ?? = [ ?? ?? ?? ?? ] and ?? ?? = [ ?? ?? ?? ?? ] then (a) ?? = ?? ?? + ?? ?? , ?? = ???? (b) ?? = ?? ?? + ?? ?? , ?? = ?? ???? (c) ?? = ?? ?? + ?? ?? , ?? = ?? ?? - ?? ?? (d) ?? = ?? ???? , ?? = ?? ?? + ?? ?? Ans: (b) ?? 2 = [ ?? ?? ?? ?? ] = [ ?? ?? ?? ?? ] [ ?? ?? ?? ?? ] = [ ?? 2 + ?? 2 2???? 2???? ?? 2 + ?? 2 ]. On comparing, we get, ?? = ?? 2 + ?? 2 , ?? = 2???? Q2. If ?? = [ ?? -?? ?? -?? ] , ?? = [ ?? ?? ?? -?? ] and (?? + ?? ) ?? = ?? ?? + ?? ?? then value of ?? and ?? are (a) ?? = ?? , ?? = ?? (b) ?? = ?? , ?? = ?? (c) ?? = ?? , ?? = ?? (d) ?? = ?? , ?? = ?? Ans: (b) We have (?? + ?? ) 2 = ?? 2 + ?? 2 + ?? · ?? + ???? ? ???? + ???? = 0 ? [ ?? - ?? 2 2?? - ?? 3 ] + [ ?? + 2 -?? - 1 ?? - 2 -?? + 1 ] = 0 [ 2?? + 2 - ?? -?? + 1 2?? - 2 4 - ?? ] = 0. On comparing, we get, - ?? + 1 = 0 ? ?? = 1 and 4 - ?? = 0 ? ?? = 4 [ 2?? + 2 - ?? -?? + 1 2?? - 2 4 - ?? ] = 0. On comparing, we get, -?? + 1 = 0 ? ?? = 1 and 4 - ?? = 0 ? ?? = 4 Q3. Let ?? (?? ) = [ ?????? ?? -?????? ?? ?? ?????? ?? ?????? ?? ?? ?? ?? ?? ]. Then ?? (?? ) · ?? (?? ' ) is equal to (a) ?? (?? ?? ' ) (b) ?? (?? /?? ' ) (c) ?? (?? + ?? ' ) (d) ?? (?? - ?? ' ) Ans: (c) We have ?? (?? ) = [ cos ?? -sin ?? 0 sin ?? cos ?? 0 0 0 1 ] , ?? (?? ' ) = [ cos ?? ' -sin ?? ' 0 sin ?? ' cos ?? ' 0 0 0 1 ] ?? (?? ) · ?? (?? ' ) = [ cos ?? -sin ?? 0 sin ?? cos ?? 0 0 0 1 ] [ cos ?? ' -sin ?? ' 0 sin ?? ' cos ?? ' 0 0 0 1 ] = [ cos (?? + ?? ' ) -sin (?? + ?? ' ) 0 sin (?? + ?? ' ) cos (?? + ?? ' ) 0 0 0 1 ] = ?? (?? + ?? ' ) Page 2 Solved Examples on Matrices JEE Mains Q1. If ?? = [ ?? ?? ?? ?? ] and ?? ?? = [ ?? ?? ?? ?? ] then (a) ?? = ?? ?? + ?? ?? , ?? = ???? (b) ?? = ?? ?? + ?? ?? , ?? = ?? ???? (c) ?? = ?? ?? + ?? ?? , ?? = ?? ?? - ?? ?? (d) ?? = ?? ???? , ?? = ?? ?? + ?? ?? Ans: (b) ?? 2 = [ ?? ?? ?? ?? ] = [ ?? ?? ?? ?? ] [ ?? ?? ?? ?? ] = [ ?? 2 + ?? 2 2???? 2???? ?? 2 + ?? 2 ]. On comparing, we get, ?? = ?? 2 + ?? 2 , ?? = 2???? Q2. If ?? = [ ?? -?? ?? -?? ] , ?? = [ ?? ?? ?? -?? ] and (?? + ?? ) ?? = ?? ?? + ?? ?? then value of ?? and ?? are (a) ?? = ?? , ?? = ?? (b) ?? = ?? , ?? = ?? (c) ?? = ?? , ?? = ?? (d) ?? = ?? , ?? = ?? Ans: (b) We have (?? + ?? ) 2 = ?? 2 + ?? 2 + ?? · ?? + ???? ? ???? + ???? = 0 ? [ ?? - ?? 2 2?? - ?? 3 ] + [ ?? + 2 -?? - 1 ?? - 2 -?? + 1 ] = 0 [ 2?? + 2 - ?? -?? + 1 2?? - 2 4 - ?? ] = 0. On comparing, we get, - ?? + 1 = 0 ? ?? = 1 and 4 - ?? = 0 ? ?? = 4 [ 2?? + 2 - ?? -?? + 1 2?? - 2 4 - ?? ] = 0. On comparing, we get, -?? + 1 = 0 ? ?? = 1 and 4 - ?? = 0 ? ?? = 4 Q3. Let ?? (?? ) = [ ?????? ?? -?????? ?? ?? ?????? ?? ?????? ?? ?? ?? ?? ?? ]. Then ?? (?? ) · ?? (?? ' ) is equal to (a) ?? (?? ?? ' ) (b) ?? (?? /?? ' ) (c) ?? (?? + ?? ' ) (d) ?? (?? - ?? ' ) Ans: (c) We have ?? (?? ) = [ cos ?? -sin ?? 0 sin ?? cos ?? 0 0 0 1 ] , ?? (?? ' ) = [ cos ?? ' -sin ?? ' 0 sin ?? ' cos ?? ' 0 0 0 1 ] ?? (?? ) · ?? (?? ' ) = [ cos ?? -sin ?? 0 sin ?? cos ?? 0 0 0 1 ] [ cos ?? ' -sin ?? ' 0 sin ?? ' cos ?? ' 0 0 0 1 ] = [ cos (?? + ?? ' ) -sin (?? + ?? ' ) 0 sin (?? + ?? ' ) cos (?? + ?? ' ) 0 0 0 1 ] = ?? (?? + ?? ' ) Q4. If ?? and ?? are square matrices of same order then (a) (???? ) ' = ?? ' ?? ' (b) (???? ) ' = ?? ' ?? ' (c) ???? = ?? , if |?? | = ?? or |?? | = ?? (d) ???? = ?? , if |?? | = ?? or ?? = ?? Ans: (b) ?? = [?? ???? ] ?? ×?? , ?? = [?? ???? ] ?? ×?? , ???? = [?? ???? ] ?? ×?? [?? ???? ] ?? ×?? = [?? ???? ] ?? ×?? , where ?? ???? = ?? ???? ?? ???? (???? ) ' = [?? ???? ] ?? ×?? ' = [?? ???? ] ?? ×?? = [?? ???? ] ?? ×?? [?? ???? ] ?? ×?? = ?? ' ?? ' Alternatively, Let ?? = [ 1 2 3 4 ] 2×2 , ?? = [ 1 3 0 4 ] 2×2 ; ???? = [ 1 11 3 25 ] (???? ) ' = [ 1 3 11 25 ] (i) and ?? ' ?? ' = [ 1 0 3 4 ] [ 1 3 2 4 ] = [ 1 3 11 25 ] From (i) and (ii), (???? ) = ?? ' ?? ' Q5. If ?? and ?? are square matrices of order ?? × ?? , then (?? - ?? ) ?? is equal to (a) ?? ?? - ?? ?? (b) ?? ?? - ?? ???? + ?? ?? (c) ?? ?? + ?? ???? + ?? ?? (d) ?? ?? - ???? - ???? + ?? ?? Ans: (d) Given ?? and ?? are square matrices of order ?? × ?? we know that (?? - ?? ) 2 = (?? - ?? )(?? - ?? ) = ?? 2 - ???? - ???? + ?? 2 Q6. Let ?? = [ ?? ?? -?? ?? -?? ?? -?? ?? ?? ], the only correct statement about the matrix ?? is (a) ?? ?? = ?? (b) ?? = (-?? )?? , where ?? is unit matrix (c) ?? -?? does not exist (d) ?? is zero matrix Ans: (a) ?? 2 = ?? · ?? = [ 0 0 -1 0 -1 0 -1 0 0 ] [ 0 0 -1 0 -1 0 -1 0 0 ] = [ 1 0 0 0 1 0 0 0 1 ] = ?? . Also, ?? -1 exists as |?? | = 1 Page 3 Solved Examples on Matrices JEE Mains Q1. If ?? = [ ?? ?? ?? ?? ] and ?? ?? = [ ?? ?? ?? ?? ] then (a) ?? = ?? ?? + ?? ?? , ?? = ???? (b) ?? = ?? ?? + ?? ?? , ?? = ?? ???? (c) ?? = ?? ?? + ?? ?? , ?? = ?? ?? - ?? ?? (d) ?? = ?? ???? , ?? = ?? ?? + ?? ?? Ans: (b) ?? 2 = [ ?? ?? ?? ?? ] = [ ?? ?? ?? ?? ] [ ?? ?? ?? ?? ] = [ ?? 2 + ?? 2 2???? 2???? ?? 2 + ?? 2 ]. On comparing, we get, ?? = ?? 2 + ?? 2 , ?? = 2???? Q2. If ?? = [ ?? -?? ?? -?? ] , ?? = [ ?? ?? ?? -?? ] and (?? + ?? ) ?? = ?? ?? + ?? ?? then value of ?? and ?? are (a) ?? = ?? , ?? = ?? (b) ?? = ?? , ?? = ?? (c) ?? = ?? , ?? = ?? (d) ?? = ?? , ?? = ?? Ans: (b) We have (?? + ?? ) 2 = ?? 2 + ?? 2 + ?? · ?? + ???? ? ???? + ???? = 0 ? [ ?? - ?? 2 2?? - ?? 3 ] + [ ?? + 2 -?? - 1 ?? - 2 -?? + 1 ] = 0 [ 2?? + 2 - ?? -?? + 1 2?? - 2 4 - ?? ] = 0. On comparing, we get, - ?? + 1 = 0 ? ?? = 1 and 4 - ?? = 0 ? ?? = 4 [ 2?? + 2 - ?? -?? + 1 2?? - 2 4 - ?? ] = 0. On comparing, we get, -?? + 1 = 0 ? ?? = 1 and 4 - ?? = 0 ? ?? = 4 Q3. Let ?? (?? ) = [ ?????? ?? -?????? ?? ?? ?????? ?? ?????? ?? ?? ?? ?? ?? ]. Then ?? (?? ) · ?? (?? ' ) is equal to (a) ?? (?? ?? ' ) (b) ?? (?? /?? ' ) (c) ?? (?? + ?? ' ) (d) ?? (?? - ?? ' ) Ans: (c) We have ?? (?? ) = [ cos ?? -sin ?? 0 sin ?? cos ?? 0 0 0 1 ] , ?? (?? ' ) = [ cos ?? ' -sin ?? ' 0 sin ?? ' cos ?? ' 0 0 0 1 ] ?? (?? ) · ?? (?? ' ) = [ cos ?? -sin ?? 0 sin ?? cos ?? 0 0 0 1 ] [ cos ?? ' -sin ?? ' 0 sin ?? ' cos ?? ' 0 0 0 1 ] = [ cos (?? + ?? ' ) -sin (?? + ?? ' ) 0 sin (?? + ?? ' ) cos (?? + ?? ' ) 0 0 0 1 ] = ?? (?? + ?? ' ) Q4. If ?? and ?? are square matrices of same order then (a) (???? ) ' = ?? ' ?? ' (b) (???? ) ' = ?? ' ?? ' (c) ???? = ?? , if |?? | = ?? or |?? | = ?? (d) ???? = ?? , if |?? | = ?? or ?? = ?? Ans: (b) ?? = [?? ???? ] ?? ×?? , ?? = [?? ???? ] ?? ×?? , ???? = [?? ???? ] ?? ×?? [?? ???? ] ?? ×?? = [?? ???? ] ?? ×?? , where ?? ???? = ?? ???? ?? ???? (???? ) ' = [?? ???? ] ?? ×?? ' = [?? ???? ] ?? ×?? = [?? ???? ] ?? ×?? [?? ???? ] ?? ×?? = ?? ' ?? ' Alternatively, Let ?? = [ 1 2 3 4 ] 2×2 , ?? = [ 1 3 0 4 ] 2×2 ; ???? = [ 1 11 3 25 ] (???? ) ' = [ 1 3 11 25 ] (i) and ?? ' ?? ' = [ 1 0 3 4 ] [ 1 3 2 4 ] = [ 1 3 11 25 ] From (i) and (ii), (???? ) = ?? ' ?? ' Q5. If ?? and ?? are square matrices of order ?? × ?? , then (?? - ?? ) ?? is equal to (a) ?? ?? - ?? ?? (b) ?? ?? - ?? ???? + ?? ?? (c) ?? ?? + ?? ???? + ?? ?? (d) ?? ?? - ???? - ???? + ?? ?? Ans: (d) Given ?? and ?? are square matrices of order ?? × ?? we know that (?? - ?? ) 2 = (?? - ?? )(?? - ?? ) = ?? 2 - ???? - ???? + ?? 2 Q6. Let ?? = [ ?? ?? -?? ?? -?? ?? -?? ?? ?? ], the only correct statement about the matrix ?? is (a) ?? ?? = ?? (b) ?? = (-?? )?? , where ?? is unit matrix (c) ?? -?? does not exist (d) ?? is zero matrix Ans: (a) ?? 2 = ?? · ?? = [ 0 0 -1 0 -1 0 -1 0 0 ] [ 0 0 -1 0 -1 0 -1 0 0 ] = [ 1 0 0 0 1 0 0 0 1 ] = ?? . Also, ?? -1 exists as |?? | = 1 Q7. 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 Q8. For how many value (?? ) of ?? in the closed interval [-?? , -?? ] is the matrix [ ?? -?? + ?? ?? ?? -?? ?? + ?? ?? + ?? -?? ?? ] singular (a) 2 (b) 0 (c) 3 (d) 2 Ans: (d) | 3 ?? - 1 2 3 -1 ?? + 2 ?? + 3 -1 2 | = 0 | 0 ?? -?? 3 -1 ?? + 2 ?? + 3 -1 2 | = 0 [?? 1 ? ?? 1 - ?? 2 ], | 0 ?? -?? -?? 0 ?? ?? + 3 -1 2 | = 0 [?? 2 ? ?? 2 - ?? 3 ] | 0 0 -?? -?? ?? ?? ?? + 3 1 2 | = 0 [?? 2 ? ?? 2 + ?? 3 ] -?? [(-?? ) - ?? (?? + 3)] = 0 ? ?? (?? 2 + 4?? ) = 0 ? ?? = 0, -4 Page 4 Solved Examples on Matrices JEE Mains Q1. If ?? = [ ?? ?? ?? ?? ] and ?? ?? = [ ?? ?? ?? ?? ] then (a) ?? = ?? ?? + ?? ?? , ?? = ???? (b) ?? = ?? ?? + ?? ?? , ?? = ?? ???? (c) ?? = ?? ?? + ?? ?? , ?? = ?? ?? - ?? ?? (d) ?? = ?? ???? , ?? = ?? ?? + ?? ?? Ans: (b) ?? 2 = [ ?? ?? ?? ?? ] = [ ?? ?? ?? ?? ] [ ?? ?? ?? ?? ] = [ ?? 2 + ?? 2 2???? 2???? ?? 2 + ?? 2 ]. On comparing, we get, ?? = ?? 2 + ?? 2 , ?? = 2???? Q2. If ?? = [ ?? -?? ?? -?? ] , ?? = [ ?? ?? ?? -?? ] and (?? + ?? ) ?? = ?? ?? + ?? ?? then value of ?? and ?? are (a) ?? = ?? , ?? = ?? (b) ?? = ?? , ?? = ?? (c) ?? = ?? , ?? = ?? (d) ?? = ?? , ?? = ?? Ans: (b) We have (?? + ?? ) 2 = ?? 2 + ?? 2 + ?? · ?? + ???? ? ???? + ???? = 0 ? [ ?? - ?? 2 2?? - ?? 3 ] + [ ?? + 2 -?? - 1 ?? - 2 -?? + 1 ] = 0 [ 2?? + 2 - ?? -?? + 1 2?? - 2 4 - ?? ] = 0. On comparing, we get, - ?? + 1 = 0 ? ?? = 1 and 4 - ?? = 0 ? ?? = 4 [ 2?? + 2 - ?? -?? + 1 2?? - 2 4 - ?? ] = 0. On comparing, we get, -?? + 1 = 0 ? ?? = 1 and 4 - ?? = 0 ? ?? = 4 Q3. Let ?? (?? ) = [ ?????? ?? -?????? ?? ?? ?????? ?? ?????? ?? ?? ?? ?? ?? ]. Then ?? (?? ) · ?? (?? ' ) is equal to (a) ?? (?? ?? ' ) (b) ?? (?? /?? ' ) (c) ?? (?? + ?? ' ) (d) ?? (?? - ?? ' ) Ans: (c) We have ?? (?? ) = [ cos ?? -sin ?? 0 sin ?? cos ?? 0 0 0 1 ] , ?? (?? ' ) = [ cos ?? ' -sin ?? ' 0 sin ?? ' cos ?? ' 0 0 0 1 ] ?? (?? ) · ?? (?? ' ) = [ cos ?? -sin ?? 0 sin ?? cos ?? 0 0 0 1 ] [ cos ?? ' -sin ?? ' 0 sin ?? ' cos ?? ' 0 0 0 1 ] = [ cos (?? + ?? ' ) -sin (?? + ?? ' ) 0 sin (?? + ?? ' ) cos (?? + ?? ' ) 0 0 0 1 ] = ?? (?? + ?? ' ) Q4. If ?? and ?? are square matrices of same order then (a) (???? ) ' = ?? ' ?? ' (b) (???? ) ' = ?? ' ?? ' (c) ???? = ?? , if |?? | = ?? or |?? | = ?? (d) ???? = ?? , if |?? | = ?? or ?? = ?? Ans: (b) ?? = [?? ???? ] ?? ×?? , ?? = [?? ???? ] ?? ×?? , ???? = [?? ???? ] ?? ×?? [?? ???? ] ?? ×?? = [?? ???? ] ?? ×?? , where ?? ???? = ?? ???? ?? ???? (???? ) ' = [?? ???? ] ?? ×?? ' = [?? ???? ] ?? ×?? = [?? ???? ] ?? ×?? [?? ???? ] ?? ×?? = ?? ' ?? ' Alternatively, Let ?? = [ 1 2 3 4 ] 2×2 , ?? = [ 1 3 0 4 ] 2×2 ; ???? = [ 1 11 3 25 ] (???? ) ' = [ 1 3 11 25 ] (i) and ?? ' ?? ' = [ 1 0 3 4 ] [ 1 3 2 4 ] = [ 1 3 11 25 ] From (i) and (ii), (???? ) = ?? ' ?? ' Q5. If ?? and ?? are square matrices of order ?? × ?? , then (?? - ?? ) ?? is equal to (a) ?? ?? - ?? ?? (b) ?? ?? - ?? ???? + ?? ?? (c) ?? ?? + ?? ???? + ?? ?? (d) ?? ?? - ???? - ???? + ?? ?? Ans: (d) Given ?? and ?? are square matrices of order ?? × ?? we know that (?? - ?? ) 2 = (?? - ?? )(?? - ?? ) = ?? 2 - ???? - ???? + ?? 2 Q6. Let ?? = [ ?? ?? -?? ?? -?? ?? -?? ?? ?? ], the only correct statement about the matrix ?? is (a) ?? ?? = ?? (b) ?? = (-?? )?? , where ?? is unit matrix (c) ?? -?? does not exist (d) ?? is zero matrix Ans: (a) ?? 2 = ?? · ?? = [ 0 0 -1 0 -1 0 -1 0 0 ] [ 0 0 -1 0 -1 0 -1 0 0 ] = [ 1 0 0 0 1 0 0 0 1 ] = ?? . Also, ?? -1 exists as |?? | = 1 Q7. 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 Q8. For how many value (?? ) of ?? in the closed interval [-?? , -?? ] is the matrix [ ?? -?? + ?? ?? ?? -?? ?? + ?? ?? + ?? -?? ?? ] singular (a) 2 (b) 0 (c) 3 (d) 2 Ans: (d) | 3 ?? - 1 2 3 -1 ?? + 2 ?? + 3 -1 2 | = 0 | 0 ?? -?? 3 -1 ?? + 2 ?? + 3 -1 2 | = 0 [?? 1 ? ?? 1 - ?? 2 ], | 0 ?? -?? -?? 0 ?? ?? + 3 -1 2 | = 0 [?? 2 ? ?? 2 - ?? 3 ] | 0 0 -?? -?? ?? ?? ?? + 3 1 2 | = 0 [?? 2 ? ?? 2 + ?? 3 ] -?? [(-?? ) - ?? (?? + 3)] = 0 ? ?? (?? 2 + 4?? ) = 0 ? ?? = 0, -4 Q9. The rank of the matrix [ -?? ?? ?? ?? -?? ?? - ?? ?? -?? ?? + ?? ] is (a) 1 if ?? = ?? (b) 2 if ?? = ?? (c) 3 if ?? = ?? (d) 1 if ?? = -?? Ans: (b,d) Let ?? = [ -1 2 5 2 -4 ?? - 4 1 -2 ?? + 1 ] = | 0 0 ?? + 6 0 0 -?? - 6 1 -2 ?? + 1 | = | 0 0 0 0 0 -?? - 6 1 -2 ?? + 1 | When ?? = -6, ?? = | 0 0 0 0 0 0 1 -2 -5 | , ? ?? (?? ) = 1 When ?? = 1, ?? = | 0 0 0 0 0 -7 1 -2 2 | ? ?? (?? ) = 2, When ?? = 6, ?? = | 0 0 0 0 0 -12 1 -2 7 | ? ?? (?? ) = 2 When ?? = 2, ?? = | 0 0 0 0 0 -8 1 -2 3 | , ? ?? (?? ) = 2 Q10. The system of linear equation ?? + ?? + ?? = ?? , ?? ?? + ?? - ?? = ?? , ?? ?? + ?? ?? + ???? = ?? has unique solution if (a) ?? ? ?? (b) -?? < ?? < ?? (c) -?? < ?? < ?? (d) ?? = ?? Ans: (a) The given system of equation has a unique solution if | 1 1 1 2 1 -1 3 2 ?? | ? 0 ? ?? ? 0 Page 5 Solved Examples on Matrices JEE Mains Q1. If ?? = [ ?? ?? ?? ?? ] and ?? ?? = [ ?? ?? ?? ?? ] then (a) ?? = ?? ?? + ?? ?? , ?? = ???? (b) ?? = ?? ?? + ?? ?? , ?? = ?? ???? (c) ?? = ?? ?? + ?? ?? , ?? = ?? ?? - ?? ?? (d) ?? = ?? ???? , ?? = ?? ?? + ?? ?? Ans: (b) ?? 2 = [ ?? ?? ?? ?? ] = [ ?? ?? ?? ?? ] [ ?? ?? ?? ?? ] = [ ?? 2 + ?? 2 2???? 2???? ?? 2 + ?? 2 ]. On comparing, we get, ?? = ?? 2 + ?? 2 , ?? = 2???? Q2. If ?? = [ ?? -?? ?? -?? ] , ?? = [ ?? ?? ?? -?? ] and (?? + ?? ) ?? = ?? ?? + ?? ?? then value of ?? and ?? are (a) ?? = ?? , ?? = ?? (b) ?? = ?? , ?? = ?? (c) ?? = ?? , ?? = ?? (d) ?? = ?? , ?? = ?? Ans: (b) We have (?? + ?? ) 2 = ?? 2 + ?? 2 + ?? · ?? + ???? ? ???? + ???? = 0 ? [ ?? - ?? 2 2?? - ?? 3 ] + [ ?? + 2 -?? - 1 ?? - 2 -?? + 1 ] = 0 [ 2?? + 2 - ?? -?? + 1 2?? - 2 4 - ?? ] = 0. On comparing, we get, - ?? + 1 = 0 ? ?? = 1 and 4 - ?? = 0 ? ?? = 4 [ 2?? + 2 - ?? -?? + 1 2?? - 2 4 - ?? ] = 0. On comparing, we get, -?? + 1 = 0 ? ?? = 1 and 4 - ?? = 0 ? ?? = 4 Q3. Let ?? (?? ) = [ ?????? ?? -?????? ?? ?? ?????? ?? ?????? ?? ?? ?? ?? ?? ]. Then ?? (?? ) · ?? (?? ' ) is equal to (a) ?? (?? ?? ' ) (b) ?? (?? /?? ' ) (c) ?? (?? + ?? ' ) (d) ?? (?? - ?? ' ) Ans: (c) We have ?? (?? ) = [ cos ?? -sin ?? 0 sin ?? cos ?? 0 0 0 1 ] , ?? (?? ' ) = [ cos ?? ' -sin ?? ' 0 sin ?? ' cos ?? ' 0 0 0 1 ] ?? (?? ) · ?? (?? ' ) = [ cos ?? -sin ?? 0 sin ?? cos ?? 0 0 0 1 ] [ cos ?? ' -sin ?? ' 0 sin ?? ' cos ?? ' 0 0 0 1 ] = [ cos (?? + ?? ' ) -sin (?? + ?? ' ) 0 sin (?? + ?? ' ) cos (?? + ?? ' ) 0 0 0 1 ] = ?? (?? + ?? ' ) Q4. If ?? and ?? are square matrices of same order then (a) (???? ) ' = ?? ' ?? ' (b) (???? ) ' = ?? ' ?? ' (c) ???? = ?? , if |?? | = ?? or |?? | = ?? (d) ???? = ?? , if |?? | = ?? or ?? = ?? Ans: (b) ?? = [?? ???? ] ?? ×?? , ?? = [?? ???? ] ?? ×?? , ???? = [?? ???? ] ?? ×?? [?? ???? ] ?? ×?? = [?? ???? ] ?? ×?? , where ?? ???? = ?? ???? ?? ???? (???? ) ' = [?? ???? ] ?? ×?? ' = [?? ???? ] ?? ×?? = [?? ???? ] ?? ×?? [?? ???? ] ?? ×?? = ?? ' ?? ' Alternatively, Let ?? = [ 1 2 3 4 ] 2×2 , ?? = [ 1 3 0 4 ] 2×2 ; ???? = [ 1 11 3 25 ] (???? ) ' = [ 1 3 11 25 ] (i) and ?? ' ?? ' = [ 1 0 3 4 ] [ 1 3 2 4 ] = [ 1 3 11 25 ] From (i) and (ii), (???? ) = ?? ' ?? ' Q5. If ?? and ?? are square matrices of order ?? × ?? , then (?? - ?? ) ?? is equal to (a) ?? ?? - ?? ?? (b) ?? ?? - ?? ???? + ?? ?? (c) ?? ?? + ?? ???? + ?? ?? (d) ?? ?? - ???? - ???? + ?? ?? Ans: (d) Given ?? and ?? are square matrices of order ?? × ?? we know that (?? - ?? ) 2 = (?? - ?? )(?? - ?? ) = ?? 2 - ???? - ???? + ?? 2 Q6. Let ?? = [ ?? ?? -?? ?? -?? ?? -?? ?? ?? ], the only correct statement about the matrix ?? is (a) ?? ?? = ?? (b) ?? = (-?? )?? , where ?? is unit matrix (c) ?? -?? does not exist (d) ?? is zero matrix Ans: (a) ?? 2 = ?? · ?? = [ 0 0 -1 0 -1 0 -1 0 0 ] [ 0 0 -1 0 -1 0 -1 0 0 ] = [ 1 0 0 0 1 0 0 0 1 ] = ?? . Also, ?? -1 exists as |?? | = 1 Q7. 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 Q8. For how many value (?? ) of ?? in the closed interval [-?? , -?? ] is the matrix [ ?? -?? + ?? ?? ?? -?? ?? + ?? ?? + ?? -?? ?? ] singular (a) 2 (b) 0 (c) 3 (d) 2 Ans: (d) | 3 ?? - 1 2 3 -1 ?? + 2 ?? + 3 -1 2 | = 0 | 0 ?? -?? 3 -1 ?? + 2 ?? + 3 -1 2 | = 0 [?? 1 ? ?? 1 - ?? 2 ], | 0 ?? -?? -?? 0 ?? ?? + 3 -1 2 | = 0 [?? 2 ? ?? 2 - ?? 3 ] | 0 0 -?? -?? ?? ?? ?? + 3 1 2 | = 0 [?? 2 ? ?? 2 + ?? 3 ] -?? [(-?? ) - ?? (?? + 3)] = 0 ? ?? (?? 2 + 4?? ) = 0 ? ?? = 0, -4 Q9. The rank of the matrix [ -?? ?? ?? ?? -?? ?? - ?? ?? -?? ?? + ?? ] is (a) 1 if ?? = ?? (b) 2 if ?? = ?? (c) 3 if ?? = ?? (d) 1 if ?? = -?? Ans: (b,d) Let ?? = [ -1 2 5 2 -4 ?? - 4 1 -2 ?? + 1 ] = | 0 0 ?? + 6 0 0 -?? - 6 1 -2 ?? + 1 | = | 0 0 0 0 0 -?? - 6 1 -2 ?? + 1 | When ?? = -6, ?? = | 0 0 0 0 0 0 1 -2 -5 | , ? ?? (?? ) = 1 When ?? = 1, ?? = | 0 0 0 0 0 -7 1 -2 2 | ? ?? (?? ) = 2, When ?? = 6, ?? = | 0 0 0 0 0 -12 1 -2 7 | ? ?? (?? ) = 2 When ?? = 2, ?? = | 0 0 0 0 0 -8 1 -2 3 | , ? ?? (?? ) = 2 Q10. The system of linear equation ?? + ?? + ?? = ?? , ?? ?? + ?? - ?? = ?? , ?? ?? + ?? ?? + ???? = ?? has unique solution if (a) ?? ? ?? (b) -?? < ?? < ?? (c) -?? < ?? < ?? (d) ?? = ?? Ans: (a) The given system of equation has a unique solution if | 1 1 1 2 1 -1 3 2 ?? | ? 0 ? ?? ? 0 Q11. The transformation due to the reflection of (?? , ?? ) through the origin is described by the matrix (a) [ ?? ?? ?? ?? ] (b) [ -?? ?? ?? -?? ] (c) [ ?? -?? -?? ?? ] (d) [ ?? ?? ?? ?? ] Ans: (b) If (?? ' , ?? ' ) is the new position ?? ' = (-1)?? + 0. ?? , ?? ' = 0. ?? + (-1)???? ? [ ?? ' ?? ' ] = [ -1 0 0 -1 ] [ ?? ?? ] ? Transformation matrix is [ -1 0 0 -1 ] Q12. The rotation through ?????? ° is identical to (a) The reflection in ?? -axis (b) The reflection in y-axis(c) A point reflection (d) Identity transformation Ans: (c) Rotation through 180 ° gives ?? ' = -?? ?? ' = -?? . Hence this a point reflection. Q13. If ?? = [ ?? -?? -?? ?? ] and ?? = [ ?? -?? -?? ?? ], then ?? ?? equals (A) ?? ?? (B) ?????? ?? (C) -?????? ?? (D) -???? ?? Ans: (B) We have ?? = ???? ? ?? 2 = (?? (?? ) 2 = ?? 2 ?? 2 = -?? 2 = - [ 2 -2 -2 2 ] = -2?? ? ?? 4 = (-2(?? ) 2 = 4?? 2 = 4(2(?? ) = 8?? ? (?? 4 ) 2 = (8(?? ) 2 ? ?? 8 = 64?? 2 = 128??Read More
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FAQs on Matrices Solved Examples - Mathematics (Maths) for JEE Main & Advanced
| 1. What are matrices in the context of JEE exam? |  |
Ans. Matrices in the context of JEE exam are rectangular arrays of numbers or symbols arranged in rows and columns. They are an important topic in mathematics and are frequently tested in JEE exams.
| 2. How are matrices used in solving problems in JEE exam? | |
Ans. Matrices are used in JEE exams to solve systems of linear equations, calculate determinants, find inverses, and perform various transformations. They are a powerful tool in solving mathematical problems efficiently.
| 3. What are the different types of matrices that are commonly tested in JEE exams? | |
Ans. The different types of matrices commonly tested in JEE exams include square matrices, symmetric matrices, skew-symmetric matrices, identity matrices, and diagonal matrices. It is important for JEE aspirants to be familiar with these different types.
| 4. How can one effectively practice solving matrix problems for JEE exam preparation? | |
Ans. To effectively practice solving matrix problems for JEE exam preparation, students should work on a variety of problems from textbooks, previous year question papers, and online resources. It is important to understand the concepts and practice regularly to improve problem-solving skills.
| 5. Are matrices an important topic for JEE exam preparation? | |
Ans. Yes, matrices are an important topic for JEE exam preparation as questions related to matrices are frequently asked in the mathematics section of the exam. It is essential for JEE aspirants to have a strong understanding of matrices to score well in the exam.
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Dec 26, 2024 Last updated |
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Additional Information about Matrices Solved Examples for JEE Preparation
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The content of Matrices Solved Examples is prepared as per the latest JEE syllabus.
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