Q.For $A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{bmatrix}$ show that $A^3 - 23A - 40I = O$, where $I$ is an identity matrix of order 3, and $O$ is zero matrix.

Matrix Equality: When Two Grids Are Truly the Same

Think of a matrix as a spreadsheet or a grid of numbers. Two matrices are equal only when they are identical in every possible way — same shape, same numbers in the same positions.

The Intuition

Imagine you have two seating charts for a classroom. Each chart is a grid where every seat has a student's name. When would you say the two charts are the same?

• First, both charts must have the same number of rows and columns — you can't compare a 3×4 chart with a 2×6 chart.
• Second, for every seat position, the name written there must match exactly between the two charts.

That's matrix equality in a nutshell.

The Precise Definition

Two matrices $A$ and $B$ are said to be equal, written $A = B$, if and only if:

1. They have the same order (same number of rows $m$ and same number of columns $n$).
2. For every position $(i, j)$, the corresponding entries are equal: $a_{ij} = b_{ij}$ for all $1 \leq i \leq m$ and $1 \leq j \leq n$.

What This Means in Practice

If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, then $A = B$ — they are the same matrix.

But if $C = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ and $D = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix}$, then $C \neq D$ because $C$ is $2 \times 3$ while $D$ is $3 \times 2$. Different shape means different matrix, even if the numbers are the same. …