Find adjoint of matrix

Engineering

Mathematics

Transpose and Adjoint of a Matrix

Inverse of a Matrix

Properties of Determinant

Question

Find adjoint of matrix $A = [\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}]$

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Let $A = [\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}]$

We know that adj A = C^T

So, let's find cofactors of each of the elements of A.

C₁₁ = (– 1)¹⁺¹ 4 = 4

C₁₂ = (– 1)¹⁺² 3 = – 3

C₂₁ = (– 1)²⁺¹ 3 = – 2

C₂₂ = (– 1)²⁺² 1 = 1

So, cofactor matrix is

C = [\begin{array}{cc} 4 & - 3 \\ - 2 & 1 \end{array}]

\Rightarrow C^{T} = [\begin{array}{cc} 4 & - 2 \\ - 3 & 1 \end{array}]

Hence,

adjA = [\begin{array}{cc} 4 & - 2 \\ - 3 & 1 \end{array}]