If A is an invertible square matrix and K is non-negative real number, then what will be the value of `(K.A)^(-1)`?

Options

A. `KA^(-1)`
B. `1/K A^(-1)`
C. `K/(A^(-1))`
D. `K^2 A^(-1)`

Answer

To find the value of \((K \cdot A)^{-1}\), where \(A\) is an invertible square matrix and \(K\) is a non-negative real number, we can follow these steps:

### Step 1: Understand the properties of matrix inversion
The inverse of a product of a scalar and a matrix can be expressed using the property of inverses. Specifically, for any scalar \(K\) and an invertible matrix \(A\), we have:
\[ (K \cdot A)^{-1} = \frac{1}{K} \cdot A^{-1} \]
This property holds true as long as \(K \neq 0\).

### Step 2: Apply the property
Since \(K\) is a non-negative real number, we need to consider two cases:
1. If \(K > 0\)
2. If \(K = 0\)

For \(K > 0\):
\[ (K \cdot A)^{-1} = \frac{1}{K} \cdot A^{-1} \]

For \(K = 0\):
\[ (0 \cdot A)^{-1} \text{ is undefined since the zero matrix does not have an inverse.} \]

### Step 3: Conclusion
Thus, we can summarize the result:
- If \(K > 0\), then \((K \cdot A)^{-1} = \frac{1}{K} \cdot A^{-1}\).
- If \(K = 0\), then \((K \cdot A)^{-1}\) is undefined.

### Final Answer
\[ (K \cdot A)^{-1} = \frac{1}{K} \cdot A^{-1} \quad \text{for } K > 0 \]
\((K \cdot A)^{-1} \text{ is undefined for } K = 0\).

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