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Matrix 2x2 (Math)

See how to solve 2x2 matrix.
17 examples and their solutions.

Adding and Subtracting Matrices

Example

A = 1234, B = 2-101
A + B = ?
Solution

Example

A = 1234, B = 2-101
A - B = ?
Solution

Example

A = 1234, B = 2-101
2A - 5B = ?
Solution

Multiplying Matrices

Example

A = 1234, B = 2-101
AB = ?
Solution

Example

A = 1234, B = 2-101
BA = ?
Solution

Zero Matrix

Definition

I = [0], 0000, 000000000, ...
A zero matrix is a matrix
whose elements are all 0.
So AO = OA = O.

Property 1

If AB = O,
then A = O or B = O. ( x )
Unlike the numbers,
AB = O doesn't mean
either A or B is a zero matrix.
(It can be a zero matrix,
but not always a zero matrix.)

Example

If AB = O, then A = O or B = O.
Counterexample?
Solution

Property 2

If AB = O,
then BA = O. ( x )

Example

If AB = O, then BA = O.
Counterexample?
Solution

Identity Matrix

Definition

AI = IA = A
The identity matrix is a matrix that satisfies
AI = IA = A.
I = [1], 1001, 100010001, ...
The identity matrix is a square matrix.
(number of rows = number of columns)
The diagonal elements are 1.
And the other elements are 0.

Example

Show that the given statement is true.
(A + I)2 = A2 + 2A + I
Solution

Cayley-Hamilton Theorem (2x2)

Theorem

A = abcd
→ A2 - (a + d)A + (ad - bc)I = O
The Cayley-Hamilton theorem can be used
to simplify An.

Example

A = 2310
Show that A3 = 7A + 6I is true.
Solution

Example

A = -1111
A10 = ?
Solution

Determinant (2x2)

Formula

abcd
→ D = ad - bc
For a 2x2 matrix,
the determinant determines
whether the inverse matrix exists.
The determinant is written as
D, det(A), abcd.
D ≠ 0 → Inverse matrix exists.
D = 0 → Inverse matrix doesn't exist.

Example

A = 1234Inverse matrix of A exists?
Solution

Inverse Matrix (2x2)

Definition

AA-1 = A-1A = I
The inverse matrix A-1 is a matrix
that satisfy this condition.
If you multiply A and A-1,
you get the identity matrix I.

Formula

A = abcd
→ A-1 = 1D d-b-ca

(D = ad - bc)
First find the determinant D.
If D ≠ 0, find A-1:
Switch a and d.
Change the signs of b and c.
If D = O,
then the inverse matrix A-1 does not exist.

Example

A = 4131
A-1 = ?
Solution

Example

A = 6834
A-1 = ?
Solution

Matrix Equation (2x2)

Formula

AX = B
→ X = A-1B
If A-1 doesn't exist,
then the matrix equation has either
infinitely many solutions
or no solution.

Example

5332 X = 8553
X = ?
Solution

System of Linear Equations: Using Matrix

Example

x - y = 4
2x + y = 5
System of Linear Equations

Solution

Example

x - y = 4
2x - 2y = 8
Solution

Example

x - y = 4
x - y = -3
Solution