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Without using the concept of inverse of a matrix, find the matrix `[x y z u]` such that `[(5,-7),(-2 ,3)][(x ,y),( z, u)]=[(-16,-6 ),(7, 2)]` .

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To solve the matrix equation \[ \begin{pmatrix} 5 & -7 \\ -2 & 3 \end{pmatrix} \begin{pmatrix} x & y \\ z & u \end{pmatrix} = \begin{pmatrix} -16 & -6 \\ 7 & 2 \end{pmatrix} \] we will follow these steps: ...
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Let A be an mxxn matrix. If there exists a matrix L of type nxxm such that LA=I_(n) , then L is called left inverse of A. Similarly, if there exists a matrix R of type nxxm such that AR=I_(m) , then R is called right inverse of A. For example, to find right inverse of matrix A=[(1,-1),(1,1),(2,3)] , we take R=[(x,y,x),(u,v,w)] and solve AR=I_(3) , i.e., [(1,-1),(1,1),(2,3)][(x,y,z),(u,v,w)]=[(1,0,0),(0,1,0),(0,0,1)] {:(implies,x-u=1,y-v=0,z-w=0),(,x+u=0,y+v=1,z+w=0),(,2x+3u=0,2y+3v=0,2z+3w=1):} As this system of equations is inconsistent, we say there is no right inverse for matrix A. The number of right inverses for the matrix [(1,-1,2),(2,-1,1)] is

Let A be an mxxn matrix. If there exists a matrix L of type nxxm such that LA=I_(n) , then L is called left inverse of A. Similarly, if there exists a matrix R of type nxxm such that AR=I_(m) , then R is called right inverse of A. For example, to find right inverse of matrix A=[(1,-1),(1,1),(2,3)] , we take R=[(x,y,x),(u,v,w)] and solve AR=I_(3) , i.e., [(1,-1),(1,1),(2,3)][(x,y,z),(u,v,w)]=[(1,0,0),(0,1,0),(0,0,1)] {:(implies,x-u=1,y-v=0,z-w=0),(,x+u=0,y+v=1,z+w=0),(,2x+3u=0,2y+3v=0,2z+3w=1):} As this system of equations is inconsistent, we say there is no right inverse for matrix A. Which of the following matrices is NOT left inverse of matrix [(1,-1),(1,1),(2,3)]?

Find the values of x,y,z from the following matrix equation : [{:(x+y,x+y+z),(2x+y,0):}]=[{:(3,6),(5,0):}]

Solve the following equations using inverse of a matrix. {:(2x+5y=1),(3x+2y=7):}

RD SHARMA-ALGEBRA OF MATRICES-Solved Examples And Exercises
  1. If A=[(1, 2 ,2 ),(2 ,1 ,2),( 2 ,2, 1)] , then prove that A^2-4A-5I=O ...

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  2. If A=[(3, 2, 0),( 1, 4 ,0 ),(0 ,0 ,5)] , show that A^2-7A+10 I3=O .

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  3. Without using the concept of inverse of a matrix, find the matrix [...

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  4. Find the matrix A such that : [(1 ,0 ),(0 ,1)]A=[(3 ,3 ,5 ),(1, 0 ,1)]

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  5. Find the matrix A such that : [2, 1, 3][(-1,0,-1),(-1,1 ,0 ),(0 ,1, 1...

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  6. Find a 2xx2 matrix A such that A[(1,-2),( 1, 4)]=6I2 .

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  7. If A=[(0, 0), (4, 0)] , find A^(16) .

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  8. If A=[(0,-x),(x,0)] , B=[(0 ,1 ),(1 ,0)] and x^2=-1 , then show tha...

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  9. If A=[(1, 0,-3 ),(2, 1 ,3 ),(0, 1 ,1)] , then verify that A^2+A=A(A+I...

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  10. If A=[(3,-5),(-4, 2)] , then find A^2-5A-14 I .

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  11. If P(x)=[cosxsinx-sinxcosx] , then show that P(x)\ P(y)=P(x+y)=P(y)\ P...

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  12. If P=[(x,0, 0),( 0,y,0 ),(0, 0,z)] and Q=[(a,0 ,0 ),(0,b,0 ),(0, 0,c)...

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  13. If A=[(2, 0, 1 ),(2, 1, 3),( 1,-1, 0)] , find A^2-5A+4I and hence fi...

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  14. If A=[(1 ,1),( 0, 1)] , prove that A^n=[(1,n),(0, 1)] for all positive...

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  15. If A=[(a, b),(0, 1)] , prove that A^n=[(a^n ,b((a^n-1)/(a-1))),(0, 1)]...

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  16. If A=[(costheta,isintheta),(isintheta,costheta)] , then prove by pri...

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  17. If A=[(cosalpha+sinalpha,sqrt(2)sinalpha),(-sqrt(2)sinalpha,cosalpha-s...

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  18. If A=[(1, 1, 1),( 0, 1, 1), (0, 0, 1)] , then use the principle of m...

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  19. If B ,\ C are n rowed square matrices and if A=B+C , B C=C B , C^2=...

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  20. If A=d i ag\ (a\ \ b\ \ c) , show that A^n=d i ag\ (a^n\ \ b^n\ \ c^n)...

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