Evaluate powers of a square matrix

How should A squared be calculated?

Powers of matrices

A matrix power is formed by multiplying a square matrix by itself repeatedly. Each multiplication uses rows against columns, so the entries are not powered one at a time

1. 1Use a square matrix A matrix can be multiplied by itself only when its number of rows equals its number of columns \(\displaystyle R_{2 \times 3}^{2}\text{ is undefined}\)
2. 2Repeat matrix multiplication The exponent tells how many copies of the same matrix are multiplied in order \(\displaystyle A^{3}=A \times A \times A\)
3. 3Use rows and columns Each entry of a power is found from a row of the left factor and a column of the right factor \(\displaystyle (A^{2})_{11}=2 \times 2+1 \times 3\)
4. 4Add the paired products A matrix power is not found by raising each entry separately \(\displaystyle \begin{bmatrix}2&1\\3&0\end{bmatrix}^{2}=\begin{bmatrix}7&2\\6&3\end{bmatrix}\)

Matrix A

\(\displaystyle A=\begin{bmatrix}2&1\\3&0\end{bmatrix}\)

For the first entry of A squared, calculate the first product from row 1 and column 1.

Matrix A

\(\displaystyle A=\begin{bmatrix}2&1\\3&0\end{bmatrix}\)

Add both products to find the entry in row 1, column 1 of A squared

Which line correctly calculates A squared?

Produce a square-matrix power with technology

Choose a matrix-capable calculator, CAS or spreadsheet. Enter the complete matrix and use actual matrix operations to calculate A squared and A cubed. Save the tool, dimensions, input order, operations and full outputs, then verify them before opening the model. This external record is self- or teacher-reviewed

1. 1Select the operation and explain it Explain why a square matrix permits repeated multiplication and why a matrix-product or matrix-power operation is appropriate. Compare your actual matrix syntax with formal A² and A³. If the tool also offers elementwise powers, record how you distinguish them. For a spreadsheet, use its full matrix-product function, not a formula that squares each cell
2. 2Construct and verify a counterexample Use technology to calculate the entrywise square as a separate operation. Compare the complete output with A² to refute the claim that matrix powers always square individual entries. Check all four entries of A² and one entry of A³ independently using row-column products
3. 3Test a special case without generalising Repeat both squaring operations for D=diag(2,3), a 2 by 2 diagonal matrix. Explain why agreement in that example does not repair the universal claim. Include the formal matrices, actual inputs, comparisons and independent checks in your record

Review matrix powers versus cellwise powers

A² has rows [7,2] and [6,3], while A³ has rows [20,7] and [21,6]. The entrywise square has rows [4,1] and [9,0], so it is not A². Your tool record must show that it actually used matrix multiplication. A* A or a power key is not sufficient evidence unless that tool treats the stored object as a matrix and the operation as matrix multiplication

1. 1Independent products Check all four entries of the square, then use the square and A to check entry (1,1) of the cube. These hand checks are separate from the saved tool outputs \(\displaystyle \begin{aligned}(A^2)_{11}&=2\times2+1\times3=7 & (A^2)_{12}&=2\times1+1\times0=2\\(A^2)_{21}&=3\times2+0\times3=6 & (A^2)_{22}&=3\times1+0\times0=3\\A^3&=(A^2)A & (A^3)_{11}&=7\times2+2\times3=20\end{aligned}\)
2. 2A genuine special case For D with diagonal entries 2,3 and off-diagonal zeros, both operations give diagonal entries 4,9 and off-diagonal zeros. Zero off-diagonal entries eliminate the cross terms. One agreeing special case cannot prove an always claim; the original A is a counterexample \(\displaystyle D=\begin{pmatrix}2&0\\0&3\end{pmatrix},\quad D^2=\begin{pmatrix}4&0\\0&9\end{pmatrix}\)
3. 3A reproducible record Record how rows and columns were entered, the exact operation names or formulas, all output entries and the row-column verification. State that these are matrix arithmetic results, not a model of an unstated practical process

Factors

\(\displaystyle A^2=\begin{pmatrix}7&2\\6&3\end{pmatrix},\quad A=\begin{pmatrix}2&1\\3&0\end{pmatrix}\)

What is entry (1,2) of A cubed?