Let A=[ 1 0 0 1 ] and B=[ a b c d ] . Show that det(A+B)=det A+det B if

step 1 Okay, in this question we want to show when determinant of a plus B is equal to determinants of

Let A=[ 1 0 0 1 ] and B=[ a b c d ]...

Let $A=\left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]$ and $B=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right] .$ Show that $\operatorname{det}(A+B)=\operatorname{det} A+\operatorname{det} B$ if and only if $a+d=0$

Key Concepts

Trace of a Matrix

The trace of a matrix is defined as the sum of its diagonal elements. For a 2x2 matrix, it is calculated as a+d, where a and d are the entries on the diagonal. In this problem, the condition a+d=0 illustrates a relationship between the trace of matrix B and the determinant of the sum A+B, demonstrating a key link between these two matrix properties.

Matrix Addition

Matrix addition is the operation of adding two matrices by adding their corresponding entries. This concept is used in the problem to form the matrix (A+B) from the identity matrix A and an arbitrary matrix B. Understanding matrix addition is essential when analyzing how the properties of matrices, such as the determinant, change under addition.

Identity Matrix

The identity matrix is a special square matrix that has ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication, meaning that any matrix multiplied by the identity matrix remains unchanged. Its determinant is always one, making it a fundamental element in matrix theory and computational proofs involving determinants.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix, such as whether it is invertible. In the context of 2x2 matrices, the determinant is calculated as ad - bc for a matrix with elements a, b, c, and d. This concept is crucial when comparing the determinant of a sum of matrices with the sum of their determinants.

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