Prove that (cA)^-1=(1/c)A^-1 If A is an invertible matrix and c is a nonzero scalar, then cA is an invertible matrix and the above equation is true. Please show step by step how you would prove it.
- print Print
- list Cite
Expert Answers
calendarEducator since 2011
write5,349 answers
starTop subjects are Math, Science, and Business
You should remember that if you need to multiply a matrix with a scalar, you need to multiply each element of the matrix with the scalar, such that:
`A = v*((a,b),(c,d)) => A = ((v*a,v*b),(v*c,v*d))`
Notice that v represents the scalar(number) and A represents the matrix.
You may prove the requested identity evaluating the inverse of the new matrix `v*A` .
You need to remember that you may evaluate the inverse of a matrix only if its determinant is not equal to zero.
Supposing that the determminant of 2x2 matrix A is not zero, yields:
`det A = va*vd - vb*vc => detA = v^2ad - v^2bc`
Factoring out `v^2` yields:
`detA = v^2(ad-bc)`
Since `det A!= 0 =>` there exists `A^(-1)` such that:
`A^(-1) = (1/(det A))((A^t_(11),A^t_(21)),(A^t_(12),A^t_(22)))`
You need to evaluate `A^t_(11)` such that:
`A^t_(11) = (-1)^(1+1)*d ` (you need to remove the terms in column 1 and row 1 and it remains the element d)
`A^t_(11) = d`
`A^t_(12) = (-1)^(1+2)*b ` (you need to remove the terms in column 1 and row 2 and it remains the element b)
`A^t_(12) = -b`
`A^t_(21) = (-1)^(2+1)*c => A^t_(21) = -c`
`A^t_(22) = (-1)^(2+2)*c => A^t_(22) = a`
`A^(-1) = (1/det A)((d,-c),(-b,a))` `=> A^(-1) = 1/(v^2(ad-bc))*((d,-c),(-b,a))`
Notice that you may write `1/det A = (1/v)*(1/(v(ad-bc)))` where `1/v` represents the inverse of the scalar v, but the next factor does not represents `1/ det A` , hence, since `A^(-1) = 1/(det A)*((d,-c),(-b,a)),` the given expression `(v*A)^(-1) = v^(-1)*A^(-1)` does not hold.
Related Questions
- Given the following matrix A, find an invertible matrix U so that A is equal to UR, when R is the...
- 1 Educator Answer
- Using properties of determinant prove that `|[1, a, a^2],[1, b, b^2],[1, c, c^2]| = (a-b)(b-c)(c-a)`
- 1 Educator Answer
- if x=a(y+z), y= b(z+x) & z= c(x+y) then, prove that, ab+bc+ca+2abc=1. no
- 1 Educator Answer
- Prove that n C r + n C r-1 = n+1 C r
- 1 Educator Answer
- Is area a vector or scalar quantity?
- 1 Educator Answer
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.