Chris 's question at Yahoo Answers (Inverse of cA)

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In summary, if A is an invertible matrix, then (cA) is also invertible and its inverse is equal to 1/c multiplied by the inverse of A. This can be shown using the properties of scalar multiplication and matrix multiplication.
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Hello Chris,

Suppose $c\neq 0$. Then, using the well known properties $(\lambda M) N=M(\lambda N)=\lambda (MN)$ and $\lambda(\mu M)=(\lambda\mu)M$ : $$(cA)\left(\frac{1}{c}A^{-1}\right)=\frac{1}{c}\left((cA)A^{-1}\right)=\frac{1}{c}\left(c\;\left(AA^{-1}\right)\right)=\left(\frac{1}{c}\cdot c\right)I=1I=I$$ So, $cA$ is invertivle and $(cA)^{-1}=\dfrac{1}{c}A^{-1}$.
 

FAQ: Chris 's question at Yahoo Answers (Inverse of cA)

What is the inverse of cA?

The inverse of cA is written as cA-1 and is the mathematical operation that undoes the effect of cA. It is also known as the multiplicative inverse or reciprocal of cA.

Why is the inverse of cA important?

The inverse of cA is important because it allows us to solve equations involving cA. It is also useful in finding the inverse of a matrix, which is crucial in many areas of science and engineering.

How do you find the inverse of cA?

To find the inverse of cA, you need to follow a specific set of steps depending on the type of cA. For example, if cA is a number, the inverse can be found by dividing 1 by cA. If cA is a matrix, there are different methods such as using the adjugate matrix or the Gauss-Jordan method.

What is the difference between the inverse of cA and the inverse function of cA?

The inverse of cA is a mathematical operation that undoes the effect of cA, while the inverse function of cA is a function that reverses the action of cA. In other words, the inverse of cA is a mathematical operation, while the inverse function of cA is a function.

Can the inverse of cA always be found?

No, the inverse of cA can only be found if cA is a non-zero number or a non-singular matrix. If cA is zero or a singular matrix, it does not have an inverse. In these cases, the operation or function involving cA is not reversible.

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