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]