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Home » Math Homework Help » Algebra Homework Help » Two Determinants Product
Two Determinants Product
Prove that the product of determinants of third order is again a determinant of the third order.

Proof: Consider the system of three linear homogeneous equations.



Now substituting the values of X, Y and Z from (2) in the system of equations given by (1) and re-arranging, we get



The system of equations given by (3) will hold only if the system of equations given by (1) holds. But the equations given by (1) will hold either if



Or X = 0, Y = 0, Z = 0.

If X = 0, Y = 0, Z = 0, then we have



Thus we conclude that system of equation (1) holds for non-zero values of x, y, z when either (4) or (5) is true. If (2) holds for non-zero values of X, Y and Z, then we must have



Thus we see that the system of equations given by (3) holds if the system of equations given by (1) holds. But the system of equations given by (1) holds for non-zero values of x, y and z when either (4) or (5) holds. Hence we conclude that (6) holds if either (4) or (5) is true.

Obviously Δ* must contain Δ and Δ1 as factors.

Let Δ* = k Δ. Δ1

Surely, both the determinants Δ* and k Δ Δ1 are of six dimensions. By comparing the coefficients of the unique term we get k = 1

∴ Δ* = Δ Δ1

Hence the product of two determinants of third order is again a determinant of third order.

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