Graded Questions on Matrices. 1. Reduce the following matrix to its normal form & hence find its rank Where
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1 Graded Questions on Matrices MATRICES Rank of Matri : Normal Form 1. Reduce the following matri to its normal form & hence find its rank A N M Q P [Dec. 07, May 10] 2. Reduce the following matri to its normal form & hence find its rank A [May 11,5marks] 3. Reduce the following matri to its normal form & hence find its rank A [Dec 10,5marks] 4. Define normal form. Reduce the following matri A to its normal form & hence find its rank A [Dec 11,6marks] 5. Reduce the following matri to its normal form & hence find its rank A N M Reduce the following matri to its normal form & hence find its rank A Q P [May 08] [Dec 03]
2 Rank of Matri : PAQ Form 1. If A N M Q P Find two non singular matrices P & Q, such that PAQ=I, where I is the unit matri and hence find A -1. [May 05, 06] 2. Find nonsingular matrices P and Q such that PAQ is in normal form. Hence Find rank of A and also find A -1 where A [May 07] 3. Find nonsingular matrices P and Q such that PAQ is in normal form. Hence Find rank of A and also find A -1 where A [May 06] 4. For the matri A, find nonsingular matrices P and Q such that PAQ is in normal form. A [May 09] 5. For the matri A, find nonsingular matrices P and Q such that PAQ is in normal form hence find rank of A where A [Dec 04] 6. Find nonsingular matrices P and Q such that PAQ is in normal form hence find A -1 if it eists A [Dec 09,6marks]
3 SYSTEM F INEAR EQUATINS 1. Solve the system of linear equations 2+y-z+3w=8 X +y +z-w=-2 3+2y-z=6 4y+3z+2w=8 [Dec 08,6marks] 2. Eamine the consistency of the consistency of the system of the following equations. If consistent, solve system of equations: X +y-z + t=2 2+3y+4t=9 Y -2z+3t=2 [Dec10, 6marks] 3. Eamine for consistency & if consistent then solve it Test for consistency & solve if consistent 2y z 3 3 y 2z 1 2 2y 3z 2 y z 1 [Dec03, 6marks] 5. Eamine for consistency & if consistent then solve it Eamine for for what value of K the equations [Dec. 2005] + y + z = y + 4z = k 4 + y + 10z = k 2 have infinite number of solutions hence find solution. INEAR DEPENDENT & INDEPENDENT 1. Test the following vectors for linear dependence and find the relation between them if dependent. [Dec 05] X 1 = (1, 2, 4), X 2 = (2, -1, 3), X 3 = (0, 1, 2), X 4 = (-3, 7, 2) 2. Eamine whether following vectors are linearly independent or dependent. X 1 = (2, 2, 1) T, X 2 = (1, 3, 1) T, X 3 = (1, 2, 2) T 3. Test the following vectors for linear dependence and find the relation between them if dependent. [Dec 03, May 10] X 1 = (3, 1,- 4), X 2 = (2, 2, -3), X 3 = (0, -4, 1) 4. Test the following vectors for linear dependence and find the relation between them if dependent. X 1 = (1, 2, -1, 0), X 2 = (1, 3, 1, 2), X 3 = (4, 2, 1, 0), X 4 = (6, 1, 0, 1)
4 5. Test the following vectors for linear dependence and find the relation between them if dependent. X 1 = (3, 1,1), X 2 = (2, 0, -1), X 3 = (4, 2, 1) 6. Test the following vectors for linear dependence and find the relation between them if dependent. [May-2014] X 1 = (1, 2, 3); X 2 = (3, -2, 1); X 3 = (1, -6, -5) 0RTHGNA MATRIX 1. Verify whether following matrices are orthogonal hence find inverse [ May-2007] A Whether following matrices are orthogonal A If not, can it be converted into an orthogonal matri 3. If [Dec- 2003, 2008] A a b c is orthogonal, find a, b, c. 4. Find l, m, n and A -1 if 0 2m n A l m n l m n is orthogonal 5. Whether following matrices are orthogonal [Dec-2005, 2007] cos 0 sin A sin 0 cos
5 6. Verify whether following matrices are orthogonal hence find inverse A EIGENVAUES AND EIGENVECTRS [Dec-2006] 1. Find the eigenvalues & eigenvectors of the matri A Find the eigenvalues & eigenvectors of the matri A Find the eigenvalues & eigenvectors of the matri A [Dec- 04] [May- 04] 4. Find the eigenvalues & eigenvectors of the matri A Find the eigenvalues & eigenvectors of the matri A Find the eigenvalues & eigenvectors of the matri [Dec-2007, May-2014] A
6 INER TRANSFRMATIN 1. Give the transformation [May-2008] Y Find the co-ordinates ( 1, 2, 3 ) corresponding to (2, 3, 0) in Y. 2. Give the transformation Y Find the co-ordinates ( 1, 2, 3 ) corresponding to (2, 9, 5) in Y. 3. Give the transformation Y Find the co-ordinates ( 1, 2, 3 ) corresponding to (3, 0, 8) in Y. 4. Give the transformation [May-2014, 2010] Y Find the co-ordinates ( 1, 2, 3 ) corresponding to (1, 2, -1) in Y. 5. Epress each of the transformation [ May-2004] 1 = 3y y 2, 2 = -y 1 + 7y 2 and y 1 = z z 2, y 2 = 4 z 1 in the matri form and find the composite transformation which epress 1, 2 in terms of z 1, z Epress each of the transformation [ May-2004] 1 = 3y y 2, 2 = -y 1 + 4y 2 and y 1 = z z 2, y 2 = 3z 1 in the matri form and find the composite transformation which epress 1, 2 in terms of z 1, z 2.
7 CAYEY-HAMITN THEREM 1. Verify Cayley-Hamilton theorem for the following matri [ May-2004, Dec-2006 ] A Verify Cayley-Hamilton theorem for the following matri [ May-2011 ] A Verify Cayley-Hamilton theorem for the following matri [May-2009 ] A Verify Cayley-Hamilton theorem for the following matri [ May-2009] A Verify Cayley-Hamilton theorem for the following matri [ May-2014 ] A Verify Cayley-Hamilton theorem for the following matri [May-2010,Dec-2004] A
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