# Distributive Properties of the Determinate

• John Creighto
In summary, the distributive property is a(b+ c)= ab+ ac. The determinant is computed by reducing one matrix to row echelon form and then transforming the product of the matrix for which the determinate is being computed.
John Creighto
I don´t have my linear algebra books with me and I forget how the distributive property of the determinate is proven. Can someone point me to a good link_

What distributive property are you talking about? The distributive property is a(b+ c)= ab+ ac. Where are you putting the determinant in that? If you are thinking "det(b+ c)= det(b)+ det(c)", that's simply not true.

HallsofIvy said:
What distributive property are you talking about? The distributive property is a(b+ c)= ab+ ac. Where are you putting the determinant in that? If you are thinking "det(b+ c)= det(b)+ det(c)", that's simply not true.

Distributive with respect to multiplication.

HallsofIvy said:
Do you mean "Det(AB)= det(A)det(B)"? That's now what I would call "distributive".

You might look at

That's what they called it at mathworld.

The definition of the determinate is:

$$a=\sum_{j_1...j_n}(-1)^ka_{1j_1}a_{2j_2}...a_{nj_n}$$

where the sum is taken over all permutations of $$\{j_1...j_n\}$$ and k=0 for even permutations and k=1 for odd permutations.

Last edited:
Yes, they do call it that! If find that very peculiar.

HallsofIvy said:
Yes, they do call it that! If find that very peculiar.

I was thinking of this argument today. Given you can compute the determinate by row operations then it seems apparent given the associativity of matrices that first reducing one matrix to reduced row echelon form via row operations and then the other via row operations;

is equivalent to taking the composition of the two matrices which reduce the matricies for which the determinate is bing computed to row echelon form and then applying this transformation to the product of the matrix product for which the determinate is being computed.

However, I was hoping for a proof which directly applied the definition of the determinate.

## What is the distributive property of the determinate?

The distributive property of the determinate is a mathematical rule that allows you to expand expressions containing parentheses. It states that when multiplying a number by a sum or difference, you can multiply each term inside the parentheses separately and then add or subtract the products to get the final result.

## How do I use the distributive property of the determinate?

To use the distributive property of the determinate, you first identify the number being multiplied by the parentheses. Then, you multiply that number by each term inside the parentheses separately. Finally, you add or subtract the resulting products to get the final answer.

## Can the distributive property of the determinate be used with variables?

Yes, the distributive property of the determinate can be used with both numbers and variables. When using it with variables, you simply treat the variable as you would any other number and multiply it by each term inside the parentheses.

## What is the difference between the distributive property of the determinate and the distributive property of multiplication?

The distributive property of the determinate is a specific rule that applies to expressions containing parentheses, while the distributive property of multiplication is a more general rule that applies to multiplying any two numbers. However, the two properties are closely related and can be used interchangeably in some cases.

## How can the distributive property of the determinate be applied in real-life situations?

The distributive property of the determinate is commonly used in algebraic equations and can be applied in various real-life situations, such as solving for unknown values in financial problems or calculating dimensions in construction projects. It is also used in computer programming and engineering to simplify complex equations and make calculations more efficient.

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