# T20 Suppose that A is a square matrix of size n and ......

• MHB
• karush
In summary: I am not to do them much any moreIn summary, the conversation discusses a problem where one needs to prove that for a square matrix A of size n and scalar $\alpha \in \CC$, $\det{\alpha A} = \alpha^n\det{A}$. The conversation mentions using induction on n to prove this, and provides an example for n=5 to demonstrate how to expand and calculate the determinant.
karush
Gold Member
MHB

$\tiny{4.288.T20}$
Suppose that A is a square matrix of size n and $\alpha \in \CC$ is $\alpha$ scalar.
Prove that $\det{\alpha A} = \alpha^n\det{A}$.
Using $\alpha=5$
$\det{5A}=\det\left(5\left[ \begin{array}{rrr} 1&2\\3&4 \end{array} \right]\right) =\det\left[ \begin{array}{rrr} 5&10\\15&20 \end{array} \right]=-50$
$5^2\det{A}=5^2\det\left[ \begin{array}{rrr} 1&2\\3&4 \end{array} \right] =\left[\begin{array}{cc} 25 & 50 \\ 75 & 100 \end{array} \right]=(25)(-50)$

Solution: $aA$ can be obtained from A by elementary row operations %of type II.
$\alpha A = E_1 \cdots E_n A$
where E, is the corresponding elementary matrix that multiplies the i-th row by the constant a.
It follows that
$\det{\alpha A}= \det{E_i}\cdots \det{E_n} \det(A)=\alpha^n\det{A}$ok I obviouly tried to follow the example above (link) but not quite sure I got the message on it...:unsure:

The problem is asking you to prove this for any positive integer, n. You cannot just show it for n= 5. I would probably use "induction on n". When n= 1 this "n by n matrix" is just a number, a. Then $det(\alpha A)= \alpha a= \alpha^1 det(A)$.

Now suppose that for A any "k by k" matrix, it is true that $det(\alpha A)= \alpha^k det(A)$ and consider B, an arbitrary k+1 by k+ 1 matrix. Calculate $det(\alpha A)$ by "expansion on the first row. You get a sum of k+ 1 terms, each a product of a number, which will be multiplied by $\alpha$, times the determinant of a k by k matrix.

Last edited:
thanks that helped a lot...

I always have a :unsure::unsure:with proofs

## 1. What is a square matrix?

A square matrix is a type of matrix where the number of rows is equal to the number of columns. It is denoted by the letter A and can be represented in a n x n format, where n is the size of the matrix.

## 2. What is the size of a square matrix?

The size of a square matrix is determined by the number of rows or columns it has. For example, a 3 x 3 square matrix has a size of 3, while a 5 x 5 square matrix has a size of 5.

## 3. What is the difference between a square matrix and a non-square matrix?

A square matrix has an equal number of rows and columns, while a non-square matrix has a different number of rows and columns. Additionally, a square matrix can have special properties and operations that are not applicable to non-square matrices.

## 4. What is the significance of a square matrix in mathematics?

Square matrices have many applications in mathematics, particularly in linear algebra. They are used to represent systems of linear equations, transformations, and eigenvalues and eigenvectors. They also have important properties such as determinants and inverses that are useful in solving various mathematical problems.

## 5. How do you perform operations on a square matrix?

Operations on a square matrix include addition, subtraction, and multiplication. These operations are performed by following specific rules and properties, such as the commutative and associative properties. In addition, square matrices can also be transformed using operations such as transposition and inversion.

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