Given:

To prove A² – 4A – 5I = 0

Here I is the identity matrix

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

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

So, Substitute corresponding values from eqn(i) in equation

A² – 4A – 5I, we get

[as r_ij = a_ij + b_ij + c_ij],

Hence the A² – 4A – 5I = 0 (Proved)