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Magazine Chapter 6: Q 6.2-63E (page 293)
Question: Show that more thann!= 1.2.3....n multiplications are required to compute the determinant of a n 脳 nmatrix by Laplace expansion (forn > 2).
Short Answer
Therefore, the determinant is given by, 
.
Step by step solution
Definition
A determinant is a unique number associated with asquare matrix.
A determinant is a scalar value that is a function of the entries of a square matrix.
It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.
To calculate the determinant
Let an be the number of multiplications required to calculate the determinant of a n 脳 n matrix.
We want to prove that 
.
Induction basis. For n = 3, we have an = 9.
Assumption,
Assume that an > n!, for an existing n > 2.
Induction step,
To calculate the determinant of a (n + 1 ) 脳 (n + 1 ) matrix, we use the Laplace expansion along any given row.
For each one of the n + 1 entries in the row, calculating the determinant of the matrix without the row and column of the entry will take more than n ! Multiplications.
So calculating the whole determinant will indeed take more than n!(n+1 ) = (n + 1 )! multiplications.
Thus, 
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