Find the matrix A such that

We know that the two matrices are eligible for their product only when the number of columns of first matrix is equal to the number of rows of the second matrix.

So, is 2×2 matrix, and is 3×2 matrix

Now in order to get a 3×2 matrix as solution 2×2 matrix should be multiplied by 2×3 matrix. Hence matrix A is 2×3 matrix.

Let,

So the given question becomes,

Now we will multiply the two matrices on LHS, we get

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

To satisfy the above equality condition, corresponding entries of the matrices should be equal, i.e.,

d = 1, e = 0, f = 1

a + d = 3 ⇒ a + 1 = 3 ⇒ a = 2

b + e = 3 ⇒ b + 0 = 3 ⇒ b = 3

c + f = 5 ⇒ c + 1 = 5 ⇒ c = 4

Now substituting these values in matrix A, we get

is the matrix A.