If find k such that A2 – 8A + kI – 0.

Q36 of 307 Page 5

If find k such that A² – 8A + kI – 0.

Given:

I is identity matrix, so

Also given,

Now, we will find the matrix for A², we get

[as c_ij = a_i1b_1j + a_i2b_2j + … + a_inb_nj]

Now, we will find the matrix for 8A, we get

So,

Substitute corresponding values from eqn(i) and (ii), we get

[as r_ij = a_ij + b_ij + c_ij],

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal

Hence,

Therefore, the value of k is 7

If show that A² –5A + 7I = 0. Use this to find A⁴.

If find k such that A² = kA – 2I₂.

If and f(x) = x² – 2x – 3, show that f(A) = 0.

If and then find λ, μ so that A² = λ A + μ I