Matrices

For the following question, enter the correct numerical value up to TWO decimal places. (If the numerical value has more than two decimal places, round-off the value to TWO decimal places.) If $A%3D%5Cbegin%7Bpmatrix%7D2%261%5C%5C%5B2pt%5D1%261%5Cend%7Bpmatrix%7D%2C%5C%3B%20B%3D%5Cbegin%7Bpmatrix%7D3%264%5C%5C%5B2pt%5D1%263%5Cend%7Bpmatrix%7D%2C%5C%3B%20C%3D%5Cbegin%7Bpmatrix%7D3%26-4%5C%5C%5B2pt%5D-2%263%5Cend%7Bpmatrix%7D%2C$ then the value of $%5Coperatorname%7Btr%7D(A)%2B%5Coperatorname%7Btr%7D%5C!%5Cleft(%5Cfrac%7BABC%7D%7B2%7D%5Cright)%20%2B%5Coperatorname%7Btr%7D%5C!%5Cleft(%5Cfrac%7BA(BC)%5E2%7D%7B4%7D%5Cright)%20%2B%5Coperatorname%7Btr%7D%5C!%5Cleft(%5Cfrac%7BA(BC)%5E3%7D%7B8%7D%5Cright)%2B%5Ccdots%20%3D%20%5C_%5C_%5C_%5C_$

Step 1: Factor the series using linearity of trace: $%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty%7D%5Coperatorname%7Btr%7D%5C!%5Cleft(A%5Cfrac%7B(BC)%5Ek%7D%7B2%5Ek%7D%5Cright)%20%3D%5Coperatorname%7Btr%7D%5C!%5Cleft(A%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty%7D%5Cleft(%5Cfrac%7BBC%7D%7B2%7D%5Cright)%5Ek%5Cright)$
Step 2: The series is a matrix geometric series: $%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty%7D%5Cleft(%5Cfrac%7BBC%7D%7B2%7D%5Cright)%5Ek%20%3D%5Cleft(I-%5Cfrac%7BBC%7D%7B2%7D%5Cright)%5E%7B-1%7D%20%5Cquad%20(%5Ctext%7Bsince%20%7D%5Crho(BC%2F2)%3C1)$
Step 3: Compute $BC$ : $BC%3D%20%5Cbegin%7Bpmatrix%7D3%264%5C%5C1%263%5Cend%7Bpmatrix%7D%20%5Cbegin%7Bpmatrix%7D3%26-4%5C%5C-2%263%5Cend%7Bpmatrix%7D%20%3D%20%5Cbegin%7Bpmatrix%7D1%260%5C%5C-3%265%5Cend%7Bpmatrix%7D$ Hence, $%5Cfrac%7BBC%7D%7B2%7D%3D%20%5Cbegin%7Bpmatrix%7D%5Ctfrac12%260%5C%5C-%5Ctfrac32%26%5Ctfrac52%5Cend%7Bpmatrix%7D$
Step 4: Compute the inverse: $I-%5Cfrac%7BBC%7D%7B2%7D%3D%20%5Cbegin%7Bpmatrix%7D%5Ctfrac12%260%5C%5C%20%5Ctfrac32%26-%5Ctfrac32%5Cend%7Bpmatrix%7D$ $%5Cleft(I-%5Cfrac%7BBC%7D%7B2%7D%5Cright)%5E%7B-1%7D%20%3D%20%5Cbegin%7Bpmatrix%7D2%260%5C%5C%202%26-%5Ctfrac23%5Cend%7Bpmatrix%7D$
Step 5: Multiply with $A$ : $A%5Cleft(I-%5Cfrac%7BBC%7D%7B2%7D%5Cright)%5E%7B-1%7D%20%3D%20%5Cbegin%7Bpmatrix%7D2%261%5C%5C1%261%5Cend%7Bpmatrix%7D%20%5Cbegin%7Bpmatrix%7D2%260%5C%5C2%26-%5Ctfrac23%5Cend%7Bpmatrix%7D%20%3D%20%5Cbegin%7Bpmatrix%7D6%26-%5Ctfrac23%5C%5C4%26-%5Ctfrac23%5Cend%7Bpmatrix%7D$
Step 6: Take trace: $%5Coperatorname%7Btr%7D%20%3D%206-%5Cfrac23%3D%5Cfrac%7B16%7D%7B3%7D%3D5.33$

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

Matrices

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

Chapter Questions
1 MCQs