Suppose A and B are two `3xx3` non singular matrices such that tr (AB) = 7.57 then tr `(BA+I_(3))`= ______

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the properties of the trace function
The trace of a matrix is the sum of its diagonal elements. Additionally, the trace has the property that for any two matrices \(X\) and \(Y\):
\[ \text{tr}(XY) = \text{tr}(YX) \]
This means that the trace of the product of two matrices is invariant under the order of multiplication.

### Step 2: Apply the properties to the given matrices
Given that \( \text{tr}(AB) = 7.57 \), we can use the property of the trace function:
\[ \text{tr}(BA) = \text{tr}(AB) = 7.57 \]

### Step 3: Calculate the trace of \(BA + I_3\)
We need to find \( \text{tr}(BA + I_3) \). Using the property of the trace that states:
\[ \text{tr}(X + Y) = \text{tr}(X) + \text{tr}(Y) \]
we can write:
\[ \text{tr}(BA + I_3) = \text{tr}(BA) + \text{tr}(I_3) \]

### Step 4: Find the trace of the identity matrix \(I_3\)
The identity matrix \(I_3\) is a \(3 \times 3\) matrix with diagonal elements all equal to 1. Therefore:
\[ \text{tr}(I_3) = 1 + 1 + 1 = 3 \]

### Step 5: Combine the results
Now we can substitute back into our equation:
\[ \text{tr}(BA + I_3) = \text{tr}(BA) + \text{tr}(I_3) = 7.57 + 3 \]
Calculating this gives:
\[ \text{tr}(BA + I_3) = 10.57 \]

### Final Answer
Thus, the value of \( \text{tr}(BA + I_3) \) is **10.57**.

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