Making a Matrix Its Own Transpose

MatricesMedium+20 XP+20

Making a Matrix Its Own Transpose

Consider

$A = \begin{bmatrix} 5 & 3x - 4 & -2 \\ x + 8 & 1 & 4y + 3 \\ -2 & 8 - y & 6 \end{bmatrix}$

Recall that the transpose is defined entry by entry as $(A^{T})_{i,j} = A_{j,i}$ .

Suppose the scalars $x$ and $y$ are chosen so that $A = A^{T}$ — transposing leaves the matrix completely unchanged.

What is $x + y$ ?

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The Transpose: Flip Rows and Columns

Drag a cell in M. Transposing doesn't just relabel — it can change how many vectors you have, and how many dimensions each one has.

M (2×3) — 3 vectors in a plane

2

3

1

2

3

Mᵀ (3×2) — 2 vectors in space

2

1

3

2

1

3

Roll

Drag to spin · use the slider to roll

M's 3 columns were 2D vectors in a plane. Mᵀ's 2 columns (M's old rows) are 3D vectors in space — transposing changed both the count and the dimension.

Making a Matrix Its Own Transpose