The matrix A<sup>2</sup> + 4A - 5I, where I is an identity matrix and A = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> equals :
32 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math>
4 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math>
4 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math>
32 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math>
Step-by-Step Solution
Key Concept: Calculate A^2, 4A, and 5I, then perform the matrix addition and subtraction: A^2 + 4A - 5I.
A = [[1, 2], [4, -3]]<br>A^2 = [[1, 2], [4, -3]] * [[1, 2], [4, -3]] = [[1+8, 2-6], [4-12, 8+9]] = [[9, -4], [-8, 17]]<br>4A = [[4, 8], [16, -12]]<br>5I = [[5, 0], [0, 5]]<br>A^2 + 4A - 5I = [[9+4-5, -4+8-0], [-8+16-0, 17-12-5]] = [[8, 4], [8, 0]] = 4 * [[2, 1], [2, 0]]
Correct Answer: 2