Matrices And determinants

Construct a matrix where Construct a matrix where Compute the value of the determinant where If A and B are two matrices such that |AB| is defined, then whether |AB|=|A||B| ?If A and B are square matrices of equal degree then which one of the following is correct?b. If A is a nn matrix, then what is |kA| ?b. If every element of a third order determinant of value D is multiplied by then what is the value of the new determinant?True or false: If every element of a determinant is equal to 1 then the value of the determinant is 1. Under what condition both AB and BA exist for two matrices A and B?If both AB and BA exist for two matrices A and B, is it necessary that AB=BA?If A is matrix and B is matrix then what is the order of AB?Find the values of and z if the following holds for matrices :Under what condition is?True or false: For two matrices A and B, .True or false: For two matrices A and B, AB = O BA= O True or False: For two square matrices A and B,.Under what condition, for any two square matrices A and B?What is the possible number of different orders which a matrix with 24 elements can have? Find the possible “square roots” of I2.True or false: If is a square matrix then is symmetric.If A is symmetric as well as skew symmetric then A is a …………….. matrix ?True or false: If A and B are symmetric matrices then AB is symmetric.Let A and B be 3×3 matrices of real numbers where A is symmetric and B is skew symmetric and. If then what are the possible values of k? (IIT 2008) If A is skew symmetric then what can you sayabout ?If A and B are skew symmetric matrices of order n what can you say about A+B, A-B, AB ?True or false: If is a square matrix then is symmetric matrix.Let be skew symmetric and is symmetric then for what value of is the matrix skew symmetric? If AB = A and BA = B then show that A and B are idempotent.Show that a square matrix A is involutary if and only if How many m×n matrices can be constructed with entries from a given set of p distinct numbers?If are aquare matrices of the same order then what is If are invertible matrices then whether are also invertible?True or False: If is invertible matrix then for any non zero scalar , .If a matrix A has its inverse equal to A itself then what is the value of A2 ?If is non singular matrix then what is the value of ?If is a non singular matrix then show that .If A= and B= then the value of for which A2 = B, is 1c. -1(IIT 2003)4d. no real valuesIf A= and |A3|= 125 then find the value of .(IIT 2004)If then c. d.Let be a square matrix all of whose entries are integers. Then which of the following are true?If then need not exist.If then exists but all its entries are not necessarily integers.If then exists and all its entries are integers.(AIEEE2008)If are matrices then = 0 implies None of theseHow many solutions does the matrix equation have?If then find If then find If commutes with thenc.d.If then show that If then show that where .If are the co factors of the corresponding elements of first row of determinant then what is the value of ?Find the value of Find the real and imaginary parts of (IIT 1997)If then the two triangles with vertices (,),(,),(,) and (,),(,),(,) are congruent. (IIT 1985)True or false: are not identically equal.(IIT 1983)Prove that (IIT 1988)=?(IIT1993)If etc and then show that If are real numbers then the determinant is independent of which quantities out of ?The parameter on which the value of the determinant does not depend upon is(IIT 1997) b. c. d.The determinant is independent of onlyc. onlyBoth d. none Show that = If are all different from and then find the value of Let a, b, c be distinct positive real numbers. Show that the value of the determinant is negative.(IIT 1981) If A= where a,b,c are real positive numbers, abc = 1 and ATA= I, then prove that .(IIT 2003)If a≠p, b≠q, c≠r and the determinant =0 then show that (IIT 1991)Find the solution set of the equation = 0(IIT 1981) then show that f (100) = 0. (IIT 1999)If Then find the domain of Solve: .(IIT 1983)If are real numbers such that Then prove that .If , then show that Solve:= 0 If then find the maximum value of the determinantIf the maximum and minimum values of are respectively then is even for any Nform the sides of a triangle. Find the distinct real roots in of = 0 (IIT 2001)The value of lying in [0, /2] such that = 0 are b. c. d.(IIT 1988)