Q.If $x, y, z$ are nonzero real numbers, then the inverse of matrix $A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ is (A) $\begin{bmatrix} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1} \end{bmatrix}$ (B) $xyz \begin{bmatrix} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1} \end{bmatrix}$ (C) $\frac{1}{xyz} \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ (D) $\frac{1}{xyz} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Concept: Inverse of a diagonal matrix — the inverse of a diagonal matrix is simply the diagonal matrix of the reciprocals of its diagonal entries (provided none are zero).

Reasoning:

1. For a diagonal matrix $A = \text{diag}(x, y, z)$, its inverse $A^{-1}$ must satisfy $A A^{-1} = I$.
2. Multiplying two diagonal matrices multiplies corresponding diagonal entries. So if $A^{-1} = \text{diag}(a, b, c)$, then $A A^{-1} = \text{diag}(xa, yb, zc)$.
3. For this to equal $I = \text{diag}(1, 1, 1)$, we need $xa = 1$, $yb = 1$, $zc = 1$. Since $x, y, z \neq 0$, we get $a = 1/x$, $b = 1/y$, $c = 1/z$. …

The key insight is that a diagonal matrix is the simplest kind of matrix to invert — because it already acts independently on each coordinate. When you multiply a diagonal matrix by a vector, each component is just scaled by its corresponding diagonal entry. To undo that scaling, you multiply by the reciprocal of that entry. So the inverse should do exactly that: scale each coordinate by $1/x$, $1/y$, and $1/z$ respectively.

Let’s verify this step by step.

1. Recall the definition of an inverse.

   A matrix $B$ is the inverse of $A$ if $AB = I$ and $BA = I$, where $I$ is the identity matrix. For a diagonal matrix, we can check this by direct multiplication.

2. Write $A$ and a candidate inverse $B$.

   Let $A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$.

   The natural guess is $B = \begin{bmatrix} 1/x & 0 & 0 \\ 0 & 1/y & 0 \\ 0 & 0 & 1/z \end{bmatrix}$, provided $x, y, z \neq 0$ (which is given).

3. Multiply $A$ by $B$.

The multiplication works because off-diagonal entries are zero — each row of $A$ has only one nonzero entry, and it lines up perfectly with the corresponding column of $B$.

4. Check $BA$ as well (though for square matrices, one-sided is enough if we know both are square).

So $B$ is indeed the inverse. …

Method: Inverting a Diagonal Matrix by Inspection

This method finds the inverse of any diagonal matrix without computing a determinant or adjoint — it applies whenever every off-diagonal entry is zero.

Steps

Step 1: Confirm the matrix is diagonal

Check that every entry off the main diagonal is $0$. If so, this shortcut applies; if not, fall back to the general adjoint method.

Step 2: Check invertibility

A diagonal matrix is invertible exactly when none of its diagonal entries is zero (since $\det D = d_1d_2\cdots d_n$, and a single zero factor makes the whole determinant zero).

Step 3: Reciprocate each diagonal entry

No cofactors, no adjoint — just invert each diagonal number in place. …

Common Mistakes

Mistake 1: Believing the inverse of a diagonal matrix involves the product $xyz$ or $1/(xyz)$

Why it's wrong: that factor belongs to the determinant and adjoint computation for a general matrix — but a diagonal matrix's inverse is found by simple reciprocation of each diagonal entry, with no determinant-style combination of the entries at all. This is exactly why options (B), (C), (D) are wrong …