Let A be a 3$\times$3 matrix with det(A)=4. Let R$_i$ denote the i$^{th}$ row of A. If a matrix B is obtained by performing the operation R$_2$ $\rightarrow$ 2R$_2$+5R$_3$ on 2A, then det(B) is equal to:
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Remember the properties of determinants under row operations:
1. R$_i$ $\leftrightarrow$ R$_j$: det changes sign.
2. R$_i$ $\rightarrow$ kR$_i$: det is multiplied by k.
3. R$_i$ $\rightarrow$ R$_i$ + kR$_j$: det is unchanged.
For a composite operation like R$_i$ $\rightarrow$ aR$_i$ + bR$_j$, the determinant is multiplied by 'a'.
We are given a 3$\times$3 matrix A with det(A) = 4.
Let C = 2A. The determinant of C is given by det(C) = det(2A).
Using the property det(kA) = k$^n$det(A) for an n$\times$n matrix:
det(C) = 2$^3$ det(A) = 8 $\times$ 4 = 32.
Now, matrix B is obtained from matrix C by the row operation R$_2$ $\rightarrow$ 2R$_2$+5R$_3$.
We can analyze the effect of this operation on the determinant in two steps:
1. The operation R$_i$ $\rightarrow$ R$_i$ + kR$_j$ does not change the value of the determinant.
2. The operation R$_i$ $\rightarrow$ kR$_i$ multiplies the determinant by k.
So, the operation R$_2$ $\rightarrow$ 2R$_2$+5R$_3$ can be thought of as first multiplying R$_2$ by 2 (which multiplies the determinant by 2) and then adding 5R$_3$ to the new R$_2$ (which does not change the determinant).
Therefore, the effect of the operation R$_2$ $\rightarrow$ 2R$_2$+5R$_3$ is to multiply the determinant by 2.
det(B) = 2 $\times$ det(C).
det(B) = 2 $\times$ 32 = 64.
