Given P = [1223] [ 1 2 2 3 ] , find P2 2 - 4P - I where I is the identity matrix

To solve the given problem, we need to follow a series of steps involving matrix operations. The original matrix **P** is given as:

P =
[ 1   2 ]
[ 2   3 ]

We are tasked with calculating **P² - 4P - I**, where **I** is the identity matrix of the same order.

Step 1: Calculate **P²**.

The square of matrix **P** is found by multiplying **P** by itself:

P² = P * P

Matrix Multiplication:
(1   2)
(2   3)
multiplied by itself gives:

P² =
[ (1 \* 1 + 2 \* 2)  (1 \* 2 + 2 \* 3) ]
[ (2 \* 1 + 3 \* 2)  (2 \* 2 + 3 \* 3) ]

P² =
[ 1 + 4    2 + 6 ]
[ 2 + 6    4 + 9 ]

P² =
[ 5  8 ]
[ 8  13 ]

Step 2: Calculate **4P**.

Multiply each element of matrix **P** by 4:

4P = 4 \*
[ 1  2 ]
[ 2  3 ]

4P =
[ 4  8 ]
[ 8  12 ]

Step 3: Identify the identity matrix **I**.

The identity matrix for a 2x2 matrix is:

I =
[ 1  0 ]
[ 0  1 ]

Step 4: Calculate **P² - 4P - I**.

Subtract 4P and I from P², element by element:

**P² - 4P - I** =
[ 5  8 ]     -
[ 4  8 ]     -
[ 1  0 ]
[ 8  13 ]   -     [ 8  12 ]   -     [ 0  1 ]

**P² - 4P - I** =
[ (5 - 4 - 1)  (8 - 8 - 0) ]
[ (8 - 8 - 0)  (13 - 12 - 1) ]

**P² - 4P - I** =
[ 0  0 ]
[ 0  0 ]

The resulting matrix is the **zero matrix**, identified as

Answer: [0000]