# Find Matrix A for 2x2 Homework Equations

• Luscinia
In summary, to find the matrix A for a 2x2, you will need to follow a specific formula where the first row contains a and b, and the second row contains c and d. The properties of a 2x2 matrix include being a rectangular array of numbers, having elements that can be real numbers, complex numbers, or variables, and following specific rules for addition, subtraction, and multiplication. To multiply two 2x2 matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The inverse of a 2x2 matrix can be found by calculating the determinant and following a specific formula. Finally, to solve a system of equations using matrix A, the equations must
Luscinia

## Homework Statement

(All matrices are 2x2. Sorry for the awkward formatting)

Find Matrix A such that
(1 -1)-1 (2 3)
(0 1 ) = (1 2) AT

## Homework Equations

A1A2...Ak=Ak-1...A2-1A1-1

## The Attempt at a Solution

(1 -1)-1 (2 3)
(0 1 ) = (1 2) AT

(1 -1)-1 (2 3)-1
(0 1 ) (1 2) = AT

((2 3)(1 -1))-1
((1 2)(0 1)) = AT

(2 1)-1
(1 1) =AT

(2 1|1 0) R1-R2 (1 0|1 -1) (1 0|1 -1)
(1 1|0 1) = (1 1|0 1) R2-R1 = (0 1|-1 2)

AT=(1 -1)
___(-1 2)
A=(1 -1)
__(-1 2)

The answer was supposed to be (2 -1)
___________________________ (-1 1)

Luscinia said:
(1 -1)-1 (2 3)
(0 1 ) = (1 2) AT

(1 -1)-1 (2 3)-1
(0 1 ) (1 2) = AT

ABA-1 ≠ B
So for the LHS, you should put $\left(^{2 3}_{1 2}\right)$-1 on the left.

## 1. How do I find the matrix A for a 2x2?

To find the matrix A for a 2x2, you will first need to understand the structure of a 2x2 matrix. It consists of 2 rows and 2 columns, and the elements of the matrix are represented by a, b, c, and d. To find the matrix A, you will need to follow the formula:

A = [a b; c d]

This means that the first row of the matrix will contain the elements a and b, and the second row will contain the elements c and d.

## 2. What are the properties of a 2x2 matrix?

There are several properties of a 2x2 matrix that are important to understand. First, the matrix is a rectangular array of numbers, arranged in rows and columns. Second, the elements of the matrix can be real numbers, complex numbers, or variables. Third, the order of the matrix is determined by the number of rows and columns, in this case 2x2. Finally, the elements of the matrix can be added, subtracted, and multiplied following specific rules.

## 3. How do I multiply two 2x2 matrices?

To multiply two 2x2 matrices, you will need to follow a specific formula. First, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Then, the product matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. The formula for multiplying two 2x2 matrices is:

[a b; c d] x [e f; g h] = [ae+bg af+bh; ce+dg cf+dh]

## 4. Can I find the inverse of a 2x2 matrix?

Yes, you can find the inverse of a 2x2 matrix. The inverse of a matrix is a matrix that when multiplied by the original matrix, results in the identity matrix. To find the inverse of a 2x2 matrix, you will first need to calculate the determinant of the matrix, which is (ad-bc). Then, you will need to follow the formula:

A^-1 = 1/(ad-bc) x [d -b; -c a]

## 5. How do I use the matrix A to solve a system of equations?

To use the matrix A to solve a system of equations, you will first need to write the system of equations in matrix form. This means that the variables will be represented as a matrix, and the constants will be represented as a separate matrix. Then, you can multiply the inverse of matrix A with the constants matrix to find the values of the variables. The formula for this is:

[x; y] = A^-1 x [c; d]

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