3.4 The inverse of a matrix (Exercises)

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To compute the inverse of a $2\times 2$-matrix.

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To compute the inverse of a $2\times 2$-matrix.

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To compute the inverse of a $2\times 2$-matrix.

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To compute the inverse of a $3\times 3$-matrix step by step.

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To compute the inverse of a $3\times 3$-matrix.

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To compute the inverse of a $3\times 3$-matrix.

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To compute the inverse of a $4\times 4$-matrix.

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To find $p$ for which a matrix $A$ is singular.

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To solve an equation $AX=B$ ($A$ and $B$ are $3\times 3$-matrices).

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True/False question about invertibility versus consistent linear systems.

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To show if $A$ is invertible, then so is $A^T$.

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To show: if $AB$ is invertible, then so are $A$ and $B$.

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What about $(AB{)}^{-1}=A^{-1}{B}^{-1}$?

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What about $((AB{)}^T{)}^{-1}=(A^T{)}^{-1}(B^T{)}^{-1}$?

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'True/False? Every elementary matrix is invertible.'

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'True/False? If $A$ and $B$ are invertible, then so is $A+B$.'

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'True/False? If $A$ and $B$ are singular, then so is $A+B$.'

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'True/False? If $A$ is row equivalent to $I$, then so is $A^2$. '

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To find 'by inspection' inverses of elemenatry matrices.

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To find the inverses of $AE$ and $EA$, when $A^{-1}$ is given.

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Finding the inverses of (almost) elementary matrices.

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Distilling $A^{-1}$ from a relation $c_2{A}^2+c_{1}A+c_{0}I=0$.

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Extricate $X$ from an equation containing $A$, $B$, $X$ and an inverse.

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Extricate $X$ from an equation containing $A$, $B$, $X$ and a transpose.

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To explore invertibility for a $3\times 2$-matrix.

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To explore invertibility for a $2\times 3$-matrix.